Area: Counting Squares and Rectangles

Find out how much flat space a shape covers. You will cover shapes with squares and count them, estimate the area of shapes with curved edges, and measure in square centimeters, square meters, square inches and square feet. Then you will see how rows and columns of squares turn into multiplication, use length × width to find the area of rectangles and squares, work backward to find a missing side, and learn how area is different from perimeter.

What you will learn

1Understand what area means
Find out that area is the amount of flat space a shape covers, and measure it by covering the shape with same-size units.
- See area as the space a shape covers
  Compare how much surface flat shapes cover, and tell area apart from length.
- Measure area by covering with units
  Cover a shape with same-size units, with no gaps and no overlaps, and count them to measure its area.
2Find area by counting squares
Count the squares inside shapes drawn on a grid, and estimate the area of shapes whose edges cut through squares.
- Count the squares inside a shape on a grid
  Count the unit squares inside a shape on squared paper carefully, without missing a square or counting one twice.
- Estimate the area of shapes with curved or uneven edges
  Estimate the area of shapes whose edges cut through squares, such as a leaf or a footprint, by counting whole squares and parts of squares.
3Measure area in standard square units
Use square centimeters, square meters, square inches and square feet, choose a sensible unit and estimate the areas of real things.
- Know the standard square units
  Learn what a square centimeter, a square meter, a square inch and a square foot are, and how to write and say them.
- Choose a unit and estimate the areas of real things
  Pick a sensible square unit for a surface, estimate its area with benchmarks, and check by measuring.
4Find the area of a rectangle by multiplying
Connect rows and columns of squares to multiplication, use length × width to find the areas of rectangles and squares, and work backward to find a missing side.
- Use rows and columns of squares
  See a rectangle covered by squares as an array, and find its area by multiplying the number of rows by the number of squares in each row.
- Use length × width to find an area
  Multiply the length of a rectangle by its width, or side by side for a square, and give the area in square units.
- Find a missing side and solve area problems
  Work backward from an area to find a missing side, find all the rectangles with a given area, and solve everyday area problems.
5Tell area and perimeter apart
Know when a problem is about the space inside a shape and when it is about the distance around it, and see that knowing one does not tell you the other.
- Find and compare the area and perimeter of a rectangle
  Find both the area and the perimeter of a rectangle, use the right units, and choose which one a real problem needs.
- Explore shapes with the same area or the same perimeter
  Find rectangles with the same area but different perimeters, and the same perimeter but different areas, and explain what this shows.

Area of a Rectangle study guide

5 chapters

How much space does a shape cover?

Think about covering your desk with paper. You need enough paper for the whole top. A long strip might reach across the desk but leave most of it bare. You will compare the space shapes cover, then measure that space with pieces that fit together.

1.1 The space inside a shape

A rug covers part of a floor. A page covers part of your desk. The amount of flat surface each shape covers is its area. A bigger area needs more material to cover it.

Compare by laying shapes together

Put a sticky note on a sheet of paper. The note fits inside, with paper still showing. The note has the smaller area. If two shapes match exactly, edge to edge, their areas are equal.

Line up the shapes carefully. A piece sticking out on one side might fit when you slide or turn it. Turning and sliding keep the area the same: you have moved the paper without adding or removing any.

Sometimes neither shape fits inside the other. A long strip may stick out past a square, while the square sticks out on both sides of the strip. Looking at just the longest side will not settle the comparison.

Length and area tell different things

Length tells how long something is. Area tells how much flat space it covers. Imagine these two paper shapes on the same grid:

Shape	Length and width	Squares covered
Thin strip	10 squares by 1 square	10
Square	4 squares by 4 squares	16

The strip is longer, but the square has more area. To compare areas, look across the whole surface, including its width.

Move the pieces

Cut a paper rectangle into two pieces. Slide the pieces into a new shape, without overlapping them. Use both pieces. The new shape has the same area as the rectangle.

Every bit of paper is still there, even though the outline changed. Overlapping pieces would hide some paper underneath. Leaving a piece out would remove some of the area you began with.

Different shapes can therefore have equal areas. They do not have to match in height, width or outline. When a direct comparison is hard, covering both shapes with the same small squares gives you a way to measure them.

Key Takeaways

Area is the amount of flat surface a shape covers.
Turning or sliding a shape keeps its area.
Rearranging all the pieces without overlaps keeps the same area.
The longest shape need not have the biggest area.

1.2 Cover with equal pieces

You can measure a book cover with playing cards. Lay the cards across it, then count them. Your count tells an area only when the pieces cover the surface properly.

Make the covering fit

Use pieces that are all the same size. Place them edge to edge across the whole shape. Leave no gaps and no overlaps.

A gap leaves surface uncovered, so your count is too small. An overlap puts two pieces over the same spot. Counting both makes the measurement too big.

Squares fit together in neat rows. Circles leave spaces between them when laid side by side. Handprints have uneven edges too, so they give only a rough measure.

Say what you counted

Suppose six playing cards cover a book exactly. Say, “The book cover has an area of 6 playing cards.” The number 6 alone leaves out the size of your measuring piece.

Now imagine someone uses smaller cards. More of those cards will fit on the same cover. The cover has not grown. The measuring unit has changed.

For a fair comparison, use the same unit on both surfaces. You cannot decide which table is bigger from “12 large sticky notes” and “30 small sticky notes.” First measure both with matching notes.

Smaller units mean a bigger count

One big square tile covers exactly as much as four small tiles. A surface takes eight big tiles. Replace each big tile with four small ones:

8 \times 4 = 32

The same surface takes 32 small tiles. Bigger units give a smaller count; smaller units give a bigger count.

Meet the unit square

A unit square has sides 1 unit long. Its area is 1 square unit. If 15 unit squares cover a shape exactly, its area is 15 square units.

People also use standard units, such as square centimeters. These have an agreed size. Your square centimeter and someone else's square centimeter are the same size, so you can compare measurements. If your counts differ, check for gaps, overlaps or a missed square.

Key Takeaways

Use same-size units across the whole surface.
Gaps make a covering count too small; overlaps make it too big.
Use the same unit when comparing two areas.
Smaller units give a larger count for the same surface.
A unit square has sides 1 unit long and area 1 square unit.

Try chapter 1:Walkthrough 6 Practice Questions 12 Flashcards

Count squares inside an outline

A grid gives you ready-made squares for measuring. Some shapes follow the grid lines, so every inside square is whole. Curved edges cross through squares and leave pieces. You will learn how to count whole squares carefully and include those pieces in an estimate.

2.1 Whole squares and half squares

On grid paper, the small squares are your measuring units. A shape's area tells how many of those squares it covers. Follow the outline so you count the inside squares.

Keep track of your count

Take a rectangle with 3 rows of 4 squares. Count across one row, then move to the next. You can put a small dot in each square as you count it.

4 + 4 + 4 = 12

The rectangle's area is 12 square units. Count the spaces inside the grid lines. Counting the lines or corners gives a different number.

This method works for shapes that are not rectangles too. Picture a plus sign with one middle square. One square joins each of its four sides. The five squares cover 5 square units. Squares outside the outline do not count, even when they touch it.

Join exact halves

A straight line from one corner of a square to the opposite corner cuts it into two equal halves. Two such half squares make one whole square.

Suppose a shape covers six whole squares and four exact half squares:

Count the six whole squares.
Pair the four halves to make two whole squares.
Add the two counts.

6 + 2 = 8

The area is exactly 8 square units. Counting each half as a whole would give too much area. Leaving all four halves out would miss part of the shape.

Draw an area yourself

Shade 10 squares that join edge to edge. You could make a long row, a rectangle or a bent shape. Each choice has an area of 10 square units because you shaded the same number of squares.

You can move the squares to change the outline. As long as you keep all 10 without overlapping them, the area stays 10 square units.

Check the unit

If each small grid square is 1 centimeter on each side, each counts as 1 cm². Your answer is then in square centimeters. If no length unit is named, write “square units.” Tracing short lengths around the outside would measure perimeter; area counts the squares inside.

Key Takeaways

Count inside squares in order, without missing or repeating any.
Two exact half squares make one whole square.
Different outlines can cover the same number of squares.
Name the square unit in your answer.

2.2 Estimate along a curved edge

Trace a leaf on grid paper. Some squares lie fully inside it. Around the edge, the leaf covers just parts of squares. You can use these pieces to find an estimate, an answer close to the real area.

Count whole squares first

Start with squares completely inside the outline. Dot each one after counting it. These squares give you a starting count, but there is still leaf area in the edge pieces.

Use this rule for the pieces: count a square as one when half or more is covered. Leave it out when less than half is covered.

A leaf covers 14 whole squares. Six more squares are more than half covered. Five are less than half covered. Add the whole squares and the six larger pieces:

14 + 6 = 20

Say, “The leaf's area is about 20 square units.” The word about matters because you judged pieces along the curve.

Put pieces together in your head

You can also imagine joining edge pieces. Two pieces that each cover about half a square make about one whole square. A large piece and a small piece may also fit together to make about one.

These curved pieces are different from exact halves cut corner to corner. You are judging their sizes, so two careful counts might differ a little. Estimates of 20 and 21 square units can both be sensible for this leaf.

Check the size of your estimate

The leaf covers all 14 whole squares and some extra surface. Its area must be more than 14 square units. It touches 25 squares altogether, but leaves some of those squares uncovered. Its area must be less than 25 square units.

Counting every touched square as a whole gives too much. Counting only the whole squares gives too little. Your estimate of 20 lies between those two limits.

A grid with smaller squares usually helps you get closer to the real area. Each edge piece you judge is smaller. Check the square size before comparing counts from two grids: more small squares need not mean more leaf.

Key Takeaways

Include edge pieces when estimating a curved shape.
One method counts half-or-more covered squares as one.
Use “about” for an estimate.
Whole squares and all touched squares give useful lower and upper limits.
Smaller grid squares usually give a closer estimate.

Try chapter 2:Walkthrough 7 Practice Questions 10 Flashcards

Give your area a unit

An area count needs a unit you can picture. Tiny squares suit a card; bigger squares suit a floor. You will read four common square units, choose one for a surface, and use familiar objects to check whether your answer makes sense.

3.1 Square centimeters, meters, inches and feet

A square unit measures an amount of flat surface. Its name tells you the side length of the square used for measuring. The surface you measure can be any shape, including a leaf or a circle.

Four useful square units

Unit	Size of one measuring square	How to write it
Square centimeter	1 centimeter on each side	cm²
Square meter	1 meter on each side	m²
Square inch	1 inch on each side	sq in or in²
Square foot	1 foot on each side	sq ft or ft²

A square centimeter is about the area of a fingernail. A square inch is about the area of a postage stamp. Many floor tiles are about one square foot. To picture a square meter, mark out a square 1 meter on each side.

What does the little ² mean?

Read cm² as “square centimeters.” The little ² names the kind of unit. It does not tell you to multiply the number before it by itself.

For example, 12 cm² means 12 squares that are each 1 cm by 1 cm. The area is twelve square centimeters. It is not 12 times 12.

A side length needs a length unit, such as cm. A covered surface needs a square unit, such as cm². On centimeter grid paper, a shape covering 18 squares has an area of 18 cm².

Look inside a large square unit

A meter is 100 centimeters long. Inside a square meter, each row holds 100 centimeter squares. There are also 100 rows:

100 \times 100 = 10,000

So 1 m² is 10,000 cm². Counting only 100 would count just one row, leaving out the other 99 rows.

A foot is 12 inches long. A square foot holds 12 rows of 12 inch squares:

12 \times 12 = 144

So 1 sq ft is 144 sq in. In both cases, the small squares fill the large square in two directions.

Key Takeaways

Square units are named by the side length of their measuring square.
The ² names a square unit; it does not square the count.
Write cm for a length and cm² for an area.
One square meter holds 10,000 square centimeters.
One square foot holds 144 square inches.
A square unit can measure a surface of any shape.

3.2 Choose a unit you can picture

Before measuring a surface, think about its size. Would small squares or big squares be easier to picture? A useful unit helps you understand the answer.

Match the unit to the surface

Use square centimeters or square inches for a stamp, card or book cover. Use square meters or square feet for a room, wall or garden.

You could measure a whole playground in square centimeters. The count would be huge and hard to picture. A single square meter would be too big for a stamp. Choose a unit that fits the job.

Start with a familiar area

A benchmark is a size you already know. These rough sizes can help:

A fingernail covers about 1 cm².
A sheet of printer paper covers about 600 cm².
A classroom door covers about 2 m².

Picture laying those areas across the surface. If about three one-meter newspaper squares would cover a whiteboard, estimate its area as about 3 m². You do not need to know the exact answer before you begin.

Measure and compare

Trace a small flat object on centimeter grid paper. Count the inside squares and include edge pieces. For a rectangle, you can measure its length and width and multiply them.

Compare your measurement with your estimate. If they are far apart, check both. Did you choose the right square unit? Did you miss a row? Did you count a square twice?

Suppose someone writes “12 cm²” for a bedroom floor. Twelve centimeter squares would cover only a tiny patch. Square meters make sense for the floor, and 12 m² is a possible bedroom area.

Check units before comparing numbers

An area of 50 cm² is much smaller than 2 m². The number 50 is bigger than 2, but the units are very different sizes. A square meter holds 10,000 square centimeters.

For a fair comparison, express both areas in the same unit. If you measured one card in square inches and another in square centimeters, measure them again with one shared unit before deciding which covers more.

Key Takeaways

Choose small square units for small surfaces and larger ones for floors or gardens.
Use a familiar area as a benchmark for an estimate.
Estimate, measure, then check a large difference.
Compare areas using the same unit.

Try chapter 3:Walkthrough 6 Practice Questions 13 Flashcards

Multiply to find a rectangle’s area

A rectangle’s squares form equal rows. That pattern lets you find the area without counting every square. You will connect the rows to side lengths, use length × width, and work backward when one side is missing. Then you can compare surfaces and plan a tile covering.

4.1 Equal rows make multiplication

A rectangle has four straight sides and four square corners. Opposite sides are equal. A square is a special rectangle with all four sides equal.

Cover a rectangle with unit squares. The squares line up in rows across and columns up and down. This arrangement is an array. Every row has the same number of squares.

Count one row, then all the rows

Imagine 4 rows with 6 squares in each row. You can add six four times:

6 + 6 + 6 + 6 = 24

Or use multiplication for those four equal groups:

4 \times 6 = 24

The area is 24 square units. You could also skip count: 6, 12, 18, 24. Each count adds one whole row.

Adding 4 and 6 would count neither all the rows nor all the squares. Multiplication works because each of the four rows contains six squares.

Turn the array

Look down the columns of that same rectangle. There are 6 columns, each holding 4 squares:

6 \times 4 = 24

Counting by columns gives the same total. Turning the rectangle switches rows and columns, while keeping every square. This helps show why changing the order of the factors keeps the product the same.

You can draw an array to show a multiplication fact. A rectangle 3 squares tall and 8 squares wide shows three groups of eight, covering 24 squares.

Fill in a partly drawn grid

You do not need to see every little square. Suppose a rectangle's top row shows 7 squares. Its side shows that it is 5 squares tall. Continue those rows and columns in your mind:

5 \times 7 = 35

The whole rectangle covers 35 square units. The top row and side tell you how to fill the inside.

Split a hard fact into easier facts

Take a rectangle 7 squares tall and 8 squares wide. Split the eight columns into five columns and three columns. Both parts are still seven rows tall.

7 \times 5 = 35

7 \times 3 = 21

35 + 21 = 56

Together, the parts cover all eight columns exactly once. So the whole rectangle has an area of 56 square units.

Key Takeaways

An array has equal rows and columns.
Multiply the number of rows by the squares in each row.
Counting by rows or columns gives the same area.
A partly drawn grid can still show the row and column counts.
Split columns into easier groups to work out a product.

4.2 Length times width

A rectangle's side lengths tell you how many unit squares fit inside it. The length gives the squares across one row. The width gives the number of rows.

Use the rectangle rule

Multiply one length by one width:

A = l \times w

Here, A means area, l means length and w means width. Choose two sides that meet at a corner. Opposite sides are equal, so using those two would repeat the same side length.

A rectangle is 8 cm long and 5 cm wide. Picture five rows of eight centimeter squares:

8 \times 5 = 40

Its area is 40 cm². Adding the four sides would measure the outside edge. Multiplying all four sides would count far too much.

No grid needed

For a rectangle labeled 9 m by 4 m, picture four rows of nine square meters:

9 \times 4 = 36

Its area is 36 m². The square unit matches the side unit. Centimeters give square centimeters; feet give square feet. A table 6 feet by 3 feet has an area of 18 square feet.

A square uses the same rule

All sides of a square are equal. Its length and width therefore have the same value. Multiply side by side. For a square with sides of 7 m:

7 \times 7 = 49

The area is 49 m². You still multiply two meeting sides.

Make the side units match

A rug is 3 m long and 200 cm wide. Change 200 cm to 2 m before multiplying:

3 \times 2 = 6

The area is 6 m². Multiplying the labels 3 and 200 directly would mix two sizes of length unit. The result 600 would not be the area in square meters or square centimeters.

An area of 1 m² need not have a square outline. A fabric strip 200 cm long and 50 cm wide covers:

200 \times 50 = 10,000

That is 10,000 cm², the same area as 1 m². The unit tells an amount of surface; it does not tell the shape of that surface.

Key Takeaways

Rectangle area is length × width, using two meeting sides.
Use the same length unit for both sides.
The area unit matches the side unit: m gives m².
Square area is side × side.

4.3 Work backward from an area

Sometimes you know how much space a rectangle covers but one side is missing. Think of its area as equal rows of squares. The unknown side is the missing number in a multiplication fact.

Find the missing width

A rectangle has an area of 24 cm² and a length of 6 cm. Ask, “Six times what makes 24?” You can divide to find out:

24 \div 6 = 4

The width is 4 cm. Check by multiplying:

6 \times 4 = 24

The answer is a side length, so write cm, not cm². Use area ÷ known side to find the other side.

For a square, look for a number that multiplies by itself. A square with area 36 m² has 6 m sides because six times six is 36.

Find different rectangles with one area

To make rectangles with area 12 square units, look for whole-number pairs that multiply to 12. These are factor pairs:

Side pair	Area
1 by 12	12 square units
2 by 6	12 square units
3 by 4	12 square units

A 4 by 3 rectangle is the 3 by 4 rectangle turned around. Try side lengths in order so you do not miss a pair.

Answer the covering problem

For a floor 8 feet by 5 feet, the area is 40 square feet. Tiles that are each 1 foot by 1 foot cover one square foot each, so you need 40 tiles for an exact covering.

Notice what the question asks for. An area needs square units. A missing side needs length units. A count of tiles needs the number of tiles.

Compare two surfaces

Room A is 6 m by 5 m, giving 30 m². Room B is 7 m by 4 m, giving 28 m². Subtract the areas:

30 - 28 = 2

Room A has 2 m² more floor space, even though Room B is longer.

Key Takeaways

Missing side = area ÷ known side.
Check a missing side by multiplying the two sides.
A side length uses cm or m, not square units.
Factor pairs give whole-number rectangles with a chosen area.
Find both areas in the same unit before comparing them.
Match your answer to the question: area, side length or tile count.

Try chapter 4:Walkthrough 14 Practice Questions 20 Flashcards

Cover the inside or measure around?

A garden needs both space for plants and a fence around its edge. Those are two different measurements. You will choose area or perimeter for each job, then try rectangles that share one measurement. Their other measurement can change when their shape changes.

5.1 Area and perimeter do different jobs

Run a finger around the outside of a picture. That distance is its perimeter. Now imagine coloring the whole picture. The amount of surface you color is its area.

Choose the measure for the job

Job	Measure	Unit examples
Put a fence around a garden	Perimeter	m, feet
Cover a floor with carpet	Area	m², square feet
Put ribbon around a card	Perimeter	cm, inches
Paint a wall	Area	m², square feet

A length unit measures distance. A square unit measures surface. The unit is part of the answer and helps show which job you worked out.

Find both for one rectangle

A rectangle is 6 cm long and 4 cm wide. For the perimeter, add all four outside sides:

6 + 4 + 6 + 4 = 20

Its perimeter is 20 cm. You can also add one length and one width, then double that sum: two lots of ten make 20.

For the area, multiply the length and width:

6 \times 4 = 24

Its area is 24 cm². These numbers measure different things. You cannot compare 24 cm² with 20 cm to decide which measurement is “bigger.”

On a grid, count the right thing

A rectangle 3 squares long and 2 squares wide covers 6 squares. Its area is 6 square units. Walk around the outside: three units across, two down, three back and two up. Its perimeter is 10 units.

All six squares touch the outside, but counting those squares would not give the perimeter. A corner square has two outside edges. Count each short outside edge when measuring the boundary.

The numbers can match

A square has sides of 4 cm. Its perimeter is 16 cm, and its area is 16 cm². Both numbers happen to be 16. You would still use one for ribbon around it and the other for paper covering it.

Before choosing a calculation, picture the job. Trace the outside for perimeter. Picture filling the surface for area.

Key Takeaways

Perimeter measures the outside distance; area measures the inside surface.
Add all four sides for a rectangle’s perimeter.
Multiply length by width for a rectangle’s area.
Use length units for perimeter and square units for area.
On a grid, count outside edge lengths for perimeter.

5.2 Same space, different fences

Two rectangles can cover the same area yet need different amounts of fence. They can also use the same amount of fence yet cover different areas. You can check both ideas with side pairs.

Keep the area the same

These rectangles each cover 12 square units:

Sides	Area	Perimeter
1 by 12	12 square units	26 units
2 by 6	12 square units	16 units
3 by 4	12 square units	14 units

Their side products match, but their side sums differ. The 3 by 4 rectangle is closest to a square and has the smallest perimeter.

You can make the long strip by cutting a 2 by 6 rectangle into two 1 by 6 strips. Put them end to end. The area stays 12 square units, while the perimeter grows from 16 to 26 units.

Keep the perimeter the same

Now use a perimeter of 20 units. Opposite sides of a rectangle are equal, so one length plus one width is half the perimeter:

20 \div 2 = 10

Find whole-number side pairs that add to 10. Then multiply each pair to find its area.

Sides	Perimeter	Area
1 by 9	20 units	9 square units
2 by 8	20 units	16 square units
3 by 7	20 units	21 square units
4 by 6	20 units	24 square units
5 by 5	20 units	25 square units

The square gives the most area. The longest, thinnest rectangle gives the least. For whole-number rectangles with the same perimeter, the sides closest in length give the biggest area.

Choose a rabbit pen

You have 16 m of fence for all four sides. Half of 16 is 8, so the length and width must add to 8 m. The choices are 1 by 7, 2 by 6, 3 by 5 and 4 by 4. Their areas are 7, 12, 15 and 16 m². Choose the 4 m by 4 m square for the most space.

A bigger area need not mean a bigger perimeter. A 1 by 9 rectangle covers 9 square units with perimeter 20 units. A 4 by 4 square covers more, 16 square units, with a smaller perimeter of 16 units.

Key Takeaways

Equal areas can have different perimeters.
Equal perimeters can enclose different areas.
For a fixed perimeter, length + width is half the perimeter.
For whole-number rectangles with equal perimeters, the closest-to-square choice has the most area.
For equal areas, the closest-to-square rectangle has the smallest perimeter.

Try chapter 5:Walkthrough 7 Practice Questions 11 Flashcards

Area of a Rectangle examples

5 Walkthroughs

Cover a surface fairly

9 steps

Compare paper shapes, then measure a covering with equal pieces.

1
IntroductionPaper over a desk
Imagine laying paper across the top of your desk. The paper covers a flat surface. The amount it covers is its area.

You will compare paper shapes, then use equal pieces to measure how much surface they cover.
2
ExampleLine up the shapes
Place a small card on a larger sheet. Slide it until none of the card sticks out. If paper still shows around it, the card has the smaller area.

If two shapes match edge to edge, their areas are equal. You are comparing the whole covered surface.
3
Key InsightA new outline
Cut a paper rectangle into two pieces. Move both pieces into a new shape, leaving no overlaps.

The outline changes, but every bit of paper is still there. Its area stays the same. Turning or sliding the whole shape also keeps its area.
4
ChallengeDoes taller mean more area?
You turn a card so it stands taller on the page. Your friend says its area has grown. Is your friend right? Explain.

---reveal---

No. Turning keeps the same paper and the same covered area. Only its direction has changed; a taller appearance does not add surface.
5
ExplanationFill every spot once
To measure area, cover the whole shape with same-size units. Fit them edge to edge.

A gap leaves surface uncounted. An overlap counts part of the surface twice. Squares fit together neatly, while circles leave spaces between them.
6
ExampleCount and name the pieces
Eight matching paper squares cover a board exactly. Each square has sides 1 unit long, so each covers 1 square unit.

The board’s area is 8 square units. The number tells how many; the unit tells the size of each measuring piece.
7
ChallengeCheck a covering
A child covers most of a card with matching square tiles but leaves a narrow gap. Does the tile count give the whole card’s area?

---reveal---

No. The gap is part of the card but is not included in the count. Fill it with the same-size units, without overlaps, to measure the whole area.
8
ChallengeTurn it and change the units
A board takes eight large square tiles. Each large tile covers four small tiles. You turn the board and cover it with the small tiles. How many do you need, and did the board’s area change?

---reveal---

You need 32 small tiles.
$8 \times 4 = 32$
Turning the board keeps its area. The count grows because the new tiles are smaller.
9
RecapWhat a covering tells you
Area measures flat surface. Turning, sliding or rearranging all the pieces without overlaps keeps it unchanged.

Measure with equal pieces that cover the whole shape. Always name the unit, especially when you compare two covering counts.

Count inside a grid outline

10 steps

Count whole and half squares, then estimate along curved edges.

Save this lesson to follow every Walkthrough step by step.

Save this lesson

1
IntroductionFollow the outline
2
ExampleOne row at a time
3
ExplanationPair the halves
4
ChallengeCount the pieces
5
ExplanationA curve leaves pieces
6
ExampleTry one edge rule
7
TipCheck both limits
8
ChallengeEstimate another outline
9
ChallengeEqual counts, different certainty
10
RecapWhole pieces and judged pieces

Choose and read a square unit

9 steps

Picture common square units and check an area against familiar sizes.

Save this lesson to follow every Walkthrough step by step.

Save this lesson

1
IntroductionGive the count a size
2
ExplanationName the measuring square
3
ExampleRead the little ²
4
ConnectionFill a bigger square
5
ChallengeSpot the unit mistake
6
ExampleUse a familiar size
7
ChallengeCheck a bedroom floor
8
ChallengeCompare with care
9
RecapPicture the unit first

From rows to missing sides

15 steps

Use arrays, side lengths and division to solve rectangle problems.

Save this lesson to follow every Walkthrough step by step.

Save this lesson

1
IntroductionA shortcut that counts every square
2
ExampleSee equal groups
3
Key InsightTurn the rows into columns
4
ExampleSplit a harder array
5
ChallengeFinish a partly drawn grid
6
ExplanationLet the sides name the rows
7
ExampleWork without a grid
8
TipMatch the units
9
ChallengeCover a square panel
10
ExampleReverse the multiplication
11
ExplanationList the side pairs
12
ChallengeFind a square side
13
ConnectionFrom area to tile count
14
ChallengeCompare after finding a side
15
RecapChoose the direction you need

Plan the inside and the edge

12 steps

Choose area or perimeter, then compare rectangles with one fixed measure.

Save this lesson to follow every Walkthrough step by step.

Save this lesson

1
IntroductionTwo jobs for one garden
2
ExplanationCover or go around?
3
ExampleOne rectangle, two answers
4
TipCount edges at a corner
5
ChallengeRibbon and paper
6
ExampleKeep the covered space
7
ExplanationKeep the fence length
8
ExampleCompare those areas
9
ChallengeChoose a garden
10
Key InsightUse the shape pattern
11
ChallengeChange the border, keep the area
12
RecapCompare the measure the job needs

Area of a Rectangle practice questions

40 Practice Questions in 4 sets

Set 1: Covering, counting and rectangles10 Practice Questions

Question 1 of 10

You turn a paper shape so its long side points upward. What happens to its area?

It gets larger.
It stays the same.
It gets smaller.
It doubles in size.

Show answerHide answer

Answer: B

Turning moves the paper without adding or removing any surface. Its area stays the same. A shape may look taller after turning, but height alone does not tell you how much space it covers.

Question 2 of 10

Eleven matching paper squares cover a tray exactly, with no gaps or overlaps. What is its area?

11 paper squares
22 paper squares
10 paper squares
12 paper squares

Show answerHide answer

Answer: A

Each paper square covers one equal part of the tray. Eleven of them cover the whole surface, so its area is 11 paper squares. Doubling the count would count the same covering twice.

Question 3 of 10

A shape covers seven whole grid squares and six exact half squares. What is its area?

13 square units
7 square units
9 square units
10 square units

Show answerHide answer

Answer: D

The six halves pair up to make three whole squares. Adding those to the seven whole squares gives 10 square units. An answer of 13 counts every half as a whole and gives too much area.

Question 4 of 10

How do you read the unit cm² aloud?

Cubic centimeters
Centimeters
Square centimeters
Two centimeters

Show answerHide answer

Answer: C

The little ² means “square,” so cm² is read as square centimeters. It names a unit of area. Reading it as centimeters leaves out the difference between a length and a covered surface.

Question 5 of 10

A rectangular grid has six rows of nine squares. How many squares does it cover?

15 squares
54 squares
30 squares
45 squares

Show answerHide answer

Answer: B

There are six equal groups of nine squares. Six times nine gives 54 squares altogether. Adding six and nine gives 15, but that does not count the squares in every row.

Question 6 of 10

A rectangular banner is 10 feet long and 9 feet wide. How much fabric covers its flat face?

38 square feet
19 square feet
81 square feet
90 square feet

Show answerHide answer

Answer: D

Multiply the two meeting side lengths: nine times ten is 90. The flat face covers 90 square feet. Adding all four sides gives 38 feet, which measures the edge and cannot tell you the amount of fabric.

Question 7 of 10

An outline covers 16 whole squares, five edge squares more than half covered, and three less than half covered. Count half-or-more covered squares as one. What area do you estimate?

21 square units
24 square units
19 square units
16 square units

Show answerHide answer

Answer: A

Count the 16 whole squares and the five larger edge pieces to get about 21 square units. Leave out the three smaller pieces under this rule. Counting all 24 touched squares as whole would include uncovered surface.

Question 8 of 10

A note says a school hallway floor has an area of 38 cm². What is wrong with this measurement?

The number should use length units.
The unit is too large for a hallway.
The area is far too small for a hallway.
The area must be written without units.

Show answerHide answer

Answer: C

Thirty-eight square centimeters would cover only a small patch of paper. A whole hallway floor is much larger, so this measurement cannot fit it. Changing to centimeters would measure a length and would not fix an area measurement.

Question 9 of 10

A rectangular display covers 42 cm² and is 7 cm long. You place a ribbon along each short side. How much ribbon do those two sides need altogether?

7 cm
12 cm
14 cm
6 cm

Show answerHide answer

Answer: B

Divide 42 by 7 to find that each short side is 6 cm. Two short sides need 12 cm of ribbon altogether. Six centimeters accounts for just one of those sides.

Question 10 of 10

You have 28 m of fence for all four sides of a rectangular pen. With whole-meter sides, what dimensions give the most space?

6 m by 8 m
5 m by 9 m
4 m by 10 m
7 m by 7 m

Show answerHide answer

Answer: D

Half of 28 is 14, so the length and width must add to 14 m. The 7 m by 7 m square covers 49 m², the most for these whole-meter choices. A 6 m by 8 m pen uses the same fence but covers only 48 m².

Set 2: Units, missing sides and borders10 Practice Questions

Save this lesson to see every answer and practice every set.

Save this lesson

Question 1 of 10

Seven large tiles cover a board exactly. Each large tile covers the same area as six small tiles. How many small tiles cover the board?

42 tiles
36 tiles
13 tiles
49 tiles

Show answerHide answer

Save this lesson to see this answer.

Question 2 of 10

A paper shape covers 19 square units. You cut it into three pieces and rearrange all of them without overlaps. What is the new area?

57 square units
22 square units
16 square units
19 square units

Show answerHide answer

Save this lesson to see this answer.

Question 3 of 10

A shape follows grid lines. It covers two squares in its top row, two in its middle row and five in its bottom row. What is its area?

7 square units
9 square units
15 square units
10 square units

Show answerHide answer

Save this lesson to see this answer.

Question 4 of 10

A curved shape covers 13 whole squares and parts of eight more squares. Between what two numbers must its area lie?

More than 8 and less than 13
More than 13 and less than 26
More than 13 and less than 21
More than 8 and less than 21

Show answerHide answer

Save this lesson to see this answer.

Question 5 of 10

A square foot has sides 12 inches long. Why does it contain 144 square inches?

Each of its four sides holds 36 squares.
It contains 144 rows of 12 squares.
Each square inch has 12-inch sides.
It contains 12 rows of 12 squares.

Show answerHide answer

Save this lesson to see this answer.

Question 6 of 10

A rectangle has nine rows of seven squares. You split it after the first five columns. How many squares are in the remaining part?

18 squares
45 squares
14 squares
28 squares

Show answerHide answer

Save this lesson to see this answer.

Question 7 of 10

A rectangular mat is 4 m long and 300 cm wide. What is its area in square meters?

1,200 m²
7 m²
12 m²
14 m²

Show answerHide answer

Save this lesson to see this answer.

Question 8 of 10

A rectangular rug has an area of 56 square feet and a length of 8 feet. What is its perimeter?

28 feet
32 feet
30 feet
15 feet

Show answerHide answer

Save this lesson to see this answer.

Question 9 of 10

A garden is 8 m by 3 m. What measurements describe its inside surface and its outside edge?

Area 22 m²; perimeter 24 m
Area 24 m²; perimeter 22 m
Area 24 m²; perimeter 11 m
Area 11 m²; perimeter 22 m

Show answerHide answer

Save this lesson to see this answer.

Question 10 of 10

You want a whole-number rectangle with area 24 square units and the shortest perimeter. What side lengths should you choose?

4 by 6
2 by 12
3 by 8
1 by 24

Show answerHide answer

Save this lesson to see this answer.

Set 3: Compare surfaces and change a design10 Practice Questions

Save this lesson to see every answer and practice every set.

Save this lesson

Question 1 of 10

A patch covers 23 cm². What does this measurement mean?

A square with sides of 23 cm
A boundary 23 cm long
An area equal to 23 centimeter squares
23 rows of 23 centimeter squares

Show answerHide answer

Save this lesson to see this answer.

Question 2 of 10

A traced shell covers 17 whole squares. Its edge pieces together cover about six half squares. What is a sensible area estimate?

20 square units
23 square units
19 square units
17 square units

Show answerHide answer

Save this lesson to see this answer.

Question 3 of 10

Nia writes “about 600 m²” for one sheet of printer paper. What change makes this a sensible area estimate?

Use about 600 cm.
Use about 6 m².
Use about 60 m².
Use about 600 cm².

Show answerHide answer

Save this lesson to see this answer.

Question 4 of 10

A rectangle has six rows of eight squares. You count five columns first. How many squares remain uncounted?

30 squares
18 squares
13 squares
3 squares

Show answerHide answer

Save this lesson to see this answer.

Question 5 of 10

A square mat is 2 feet long and 24 inches wide. What is its area in square feet?

48 square feet
4 square feet
24 square feet
8 square feet

Show answerHide answer

Save this lesson to see this answer.

Question 6 of 10

You found rectangles 1 by 30, 2 by 15 and 3 by 10 with area 30 square units. What whole-number side pair is still missing, counting turned rectangles once?

4 by 7
6 by 6
5 by 5
5 by 6

Show answerHide answer

Save this lesson to see this answer.

Question 7 of 10

A square notice has a perimeter of 24 cm. You make two copies of this notice at the same size. How much paper covers both copies altogether?

72 cm²
48 cm²
144 cm²
36 cm²

Show answerHide answer

Save this lesson to see this answer.

Question 8 of 10

A 2 m by 10 m garden is changed to a 4 m by 5 m garden. How much less fence is needed around all four sides?

8 m less
3 m less
6 m less
2 m less

Show answerHide answer

Save this lesson to see this answer.

Question 9 of 10

Sixteen large square tiles cover one table exactly. Each large tile can be replaced by four small tiles. You want to cover two identical tables and have 60 small tiles. How many more small tiles do you need?

68 tiles
4 tiles
128 tiles
64 tiles

Show answerHide answer

Save this lesson to see this answer.

Question 10 of 10

A rectangular patio covers 63 m² and is 9 m long. Its width grows by 2 m, while its length stays the same. What is its new area?

72 m²
79 m²
77 m²
81 m²

Show answerHide answer

Save this lesson to see this answer.

Set 4: Plan coverings, edges and changes10 Practice Questions

Save this lesson to see every answer and practice every set.

Save this lesson

Question 1 of 10

A paper strip is 11 squares by 2 squares. A paper rectangle is 6 squares by 4 squares on the same grid. Which covers more, and by how much?

The strip, by 2 square units.
The rectangle, by 4 square units.
The strip, by 5 square units.
The rectangle, by 2 square units.

Show answerHide answer

Save this lesson to see this answer.

Question 2 of 10

A sticker covers 45 cm². A small rug covers 3 m². Which covers more surface?

The sticker, because 45 is more than 3.
The rug, because each m² is much larger.
They cover equal amounts of surface.
The sticker, because its unit is smaller.

Show answerHide answer

Save this lesson to see this answer.

Question 3 of 10

A rectangular wall is 6 m long and 200 cm tall. What area will a mural cover on the whole wall?

12 m²
8 m²
16 m²
1,200 m²

Show answerHide answer

Save this lesson to see this answer.

Question 4 of 10

A grid shape covers 15 whole squares and ten exact half squares. What is its area?

25 square units
19 square units
20 square units
15 square units

Show answerHide answer

Save this lesson to see this answer.

Question 5 of 10

A rectangle contains five rows of nine unit squares. What is its area?

14 square units
28 square units
45 square units
36 square units

Show answerHide answer

Save this lesson to see this answer.

Question 6 of 10

A rectangular play mat covers 72 square feet and is 8 feet long. You trim its length to 7 feet and keep its width. What area remains?

63 square feet
65 square feet
64 square feet
62 square feet

Show answerHide answer

Save this lesson to see this answer.

Question 7 of 10

A rectangular picture has a perimeter of 26 cm and a length of 8 cm. What is its area?

44 cm²
48 cm²
36 cm²
40 cm²

Show answerHide answer

Save this lesson to see this answer.

Question 8 of 10

A club has 30 m of fence for a rectangular garden with whole-meter sides. What is the best design, and how much more area does it give than a 6 m by 9 m garden?

8 m by 8 m; 10 m² more
7 m by 8 m; 2 m² more
6 m by 9 m; no more
7 m by 8 m; 1 m² more

Show answerHide answer

Save this lesson to see this answer.

Question 9 of 10

Two leaves are traced on the same grid. Leaf A covers 15 whole squares, four more-than-half pieces and six smaller pieces. Leaf B covers 13 whole squares, seven more-than-half pieces and three smaller pieces. Count half-or-more pieces as one. How do the estimates compare?

Leaf B is about 1 square unit larger.
Leaf A is about 2 square units larger.
Leaf A is about 3 square units larger.
Leaf B is about 4 square units larger.

Show answerHide answer

Save this lesson to see this answer.

Question 10 of 10

A 2 m by 14 m garden will be reshaped with the same area and the shortest possible perimeter. With whole-meter sides, how much fence will you save?

8 m
5 m
10 m
6 m

Show answerHide answer

Save this lesson to see this answer.

Stuck on your own question? Homework Help

Key Terms

42 Key Terms, each with its definition

A

Area

The area of a shape is the amount of flat surface it covers.

You find a rug’s area to know how much floor it covers.

Area formula

An area formula is a rule for calculating how much surface a shape covers.

The rectangle’s area formula is length times width.

Array

An array is an arrangement of objects or squares in equal rows and columns.

A rectangular array shows a multiplication fact as groups of equal size.

B

Benchmark

A benchmark is a familiar size you use to estimate or check another measurement.

A sheet of printer paper is an area benchmark of about 600 cm².

Boundary

A boundary is the outside edge that separates the inside of a shape from the space around it.

Trace the boundary of a card to measure the length of ribbon it needs.

C

Closest-to-square rectangle

The closest-to-square rectangle has the most nearly equal sides among rectangles with the same area or perimeter.

For whole-number sides and a fixed perimeter, the closest-to-square choice has the most area.

Column

A column is a line of objects or squares going up and down an arrangement.

Counting the squares in each column gives the same total as counting by rows.

Customary area unit

A customary area unit measures surface using customary lengths, such as inches or feet.

Square inches are a useful customary area unit for a small label.

E

Equal areas

Shapes have equal areas when they cover the same amount of surface, even with different outlines.

A thin rectangle and a wider rectangle can have equal areas but different perimeters.

Equal perimeters

Shapes have equal perimeters when the distances around their outside edges are the same.

Two gardens with equal perimeters can leave different amounts of space for plants.

Estimate

An estimate is a sensible answer close to the real amount, found without measuring it exactly.

Judging pieces along a leaf’s edge gives an estimate of its area.

F

Factor pair

A factor pair is two whole numbers that multiply to give a chosen number.

The pair 3 and 4 gives a rectangle with area 12 square units.

Fixed perimeter

A fixed perimeter is an outside distance that stays the same while you try different shapes.

Using all 20 m of fence each time keeps a garden’s perimeter fixed at 20 m.

G

Gap

A gap is an uncovered space between pieces used to measure an area.

A gap between two tiles leaves some of the board uncounted.

Grid

A grid is a pattern of crossing lines that makes rows and columns of spaces.

You can trace a card on a square grid and count the spaces inside its outline.

H

Half square

A half square is one of two equal pieces of a square. Two exact halves make one whole.

A diagonal from corner to corner cuts a grid square into two half squares.

L

Length

A length measures how far something goes from one end to another.

A long, narrow strip can have more length but less area than a wide card.

Length unit

A length unit measures distance in one direction, such as a centimeter, meter, inch or foot.

Use a length unit for perimeter and a square unit for area.

Lower limit

A lower limit is a value below which an amount cannot go.

The whole squares give a lower limit for a leaf’s area; its edge pieces add more.

M

Measuring unit

A measuring unit is a known amount you use repeatedly to measure something.

Matching sticky notes can serve as area units when they cover a surface exactly.

Meeting sides

Meeting sides are two sides that join at a corner of a shape.

Choose two meeting sides for the rectangle’s length and width.

Metric area unit

A metric area unit measures surface using metric lengths, such as centimeters or meters.

You might choose square meters as the metric area unit for a floor.

Missing factor

A missing factor is an unknown number in a multiplication with a known product.

In “6 times what makes 24?”, the missing factor gives the unknown rectangle side.

Missing side

A missing side is a side whose length you need to find from other measurements.

Divide a rectangle’s area by its known side to find the missing side.

O

Overlap

An overlap happens when part of one covering piece lies on top of another.

Two overlapping cards count some of the book cover’s surface twice.

P

Part square

A part square is the portion of a grid square covered where an outline crosses through it.

You include part squares when estimating the area inside a curved outline.

Perimeter

The perimeter of a shape is the distance all the way around its outside edge.

A fence around all four sides of a garden follows its perimeter.

R

Rectangle

A rectangle is a flat shape with four straight sides and four square corners. Its opposite sides are equal.

A book cover can be a rectangle, with equal top and bottom edges.

Repeated addition

Repeated addition means adding the same number several times to count equal groups.

Adding six four times counts four rows of six squares.

Row

A row is a line of objects or squares going across an arrangement.

Four equal rows of six squares contain 24 squares altogether.

S

Square

A square is a rectangle with all four sides equal in length.

A square tile with 3 cm sides covers 9 cm².

Square centimeter

A square centimeter is the area of a square with sides 1 centimeter long, written cm².

A fingernail covers about one square centimeter of surface.

Square foot

A square foot is the area of a square with sides 1 foot long, written sq ft or ft².

A square foot holds 12 rows of 12 inch squares, giving 144 square inches.

Square inch

A square inch is the area of a square with sides 1 inch long, written sq in or in².

A postage stamp covers roughly one square inch of surface.

Square meter

A square meter is the area of a square with sides 1 meter long, written m².

One square meter holds 100 rows of 100 centimeter squares.

Square unit

A square unit is an area equal to a square whose sides are each 1 unit long.

A shape covering nine unit squares has an area of 9 square units.

Standard unit

A standard unit is a measuring unit with a size that people have agreed to use.

Square centimeters let two people measure with matching sizes of square.

Surface

A surface is the outside face of an object, such as the flat top of a desk.

Paper laid across the desk covers part of its top surface.

U

Unit square

A unit square is a square with sides 1 unit long and an area of 1 square unit.

Fifteen unit squares cover a shape with area 15 square units.

Upper limit

An upper limit is a value above which an amount cannot go.

Counting every square a leaf touches gives an upper limit for its area.

W

Whole square

A whole square in an area count is a grid square completely covered by the shape.

You count each whole square inside an outline as one square unit.

Width

The width of a rectangle is the side length across it, meeting its length at a corner.

Multiply a rectangle’s length by its width to find its area.

About this lesson

By StudeliPublished November 13, 2026Report a mistake

Keep learning

More courses

Make your own Study Materials

Turn your notes, slides and documents into Study Guides, Practice Questions, Flashcards and 6 more tools.

Get Started