Area of Triangles, Parallelograms and Trapezoids

A floor needs tiles across its whole surface. A fence follows the edge of a yard. These jobs need different measurements: area for the surface and perimeter for the edge.

Count squares with multiplication

A unit square has sides one unit long. Cover a shape with these squares, with no gaps or overlaps. The number of squares is its area.

A rectangle 6 cm long and 4 cm wide holds four rows. Each row has six square centimeters:

Read this as “24 square centimeters.” The small 2 shows that the unit is a square. Its perimeter is a length:

For any rectangle, $A=lw$ means length times width. You can also write $A=bh$, or base times height. The height meets the base at a right angle. Rectangles will help explain the area rules for other shapes.

Choose units that match

Small surfaces may use square centimeters or square inches. Larger surfaces may use square feet, square yards or square meters. On a grid with no named unit, write “square units.”

Before multiplying lengths, put them in the same unit. A rectangle 2 m long and 50 cm wide can be measured in meters:

Or change 2 m to 200 cm:

Both answers describe the same surface. Multiplying 2 by 50 mixes two different length units.

Convert the whole square

One foot is 12 inches. A square foot has 12 rows of 12 square inches:

The length conversion works in both directions across the square. In the same way:

The area rules also work with fractions and decimals. A rectangle 2½ ft by 4 ft has area 10 ft². A rectangle 3.5 m by 1.2 m has area 4.2 m².

Cut and move pieces

Cutting a paper shape does not remove any paper. Move the pieces into a new shape without overlaps. Their total area stays the same. Also, adding the areas of pieces that exactly fill a shape gives its whole area. These facts let you turn unfamiliar shapes into ones you can measure.

The base of a triangle does not have to be at the bottom. Turn the drawing and a different side may be lowest. The shape still covers the same area.

Choose a side, then find its height

A base is a side you choose to measure from. Its matching height is measured at a right angle to that side. A right angle measures 90°. Lines that meet at this angle are perpendicular.

In a triangle, draw the height from the opposite corner to the base line. A corner is also called a vertex. In a parallelogram, measure straight across between the chosen base and its opposite parallel side.

A dashed line often shows the height. A small square at its foot marks the right angle. Follow that mark when choosing lengths. A slanted side that does not meet the base at a right angle cannot be its height.

Every base has its own height. A triangle has three possible base–height pairs. A parallelogram has a pair for each direction of its sides. Changing the base means finding the height that belongs to the new base.

Find heights inside and outside

In a right triangle, the two sides meeting at the right angle are the legs. Choose either leg as the base; the other leg is its height. The longest side is opposite the right angle. It does not form a right angle with either leg.

For an obtuse triangle, one angle is greater than 90°. Some heights lie outside the triangle. Extend the base line with a dashed line. From the opposite corner, draw a perpendicular to that line.

The extra dashed length helps locate the height. It does not make the triangle's base longer or add any surface. Use the length of the actual side as the base.

For an acute triangle, all angles are less than 90°. Each height lies inside the triangle. “Altitude” is another name for a triangle's height.

Read heights on a grid

Suppose a triangle's horizontal base is six squares long. Its opposite corner is four rows above the base line. The height is four units, even when that corner sits beyond the end of the base. Count rows straight across the gap; do not count along the leaning edge.

A useful length check comes from the shortest route. The perpendicular height is the shortest distance to the base line. In a parallelogram with sides 8 cm and 5 cm, the height to the 8 cm base cannot exceed 5 cm. It reaches 5 cm when the shape is a rectangle. A label showing a height of 6 cm beside that 5 cm side would be impossible.

A parallelogram has two pairs of parallel sides. Its opposite sides are equal, and its opposite angles are equal. Rectangles, squares and rhombuses belong to this family.

Slide a triangle to the other end

Imagine a paper parallelogram leaning to the right. Choose a longer side as the base. Draw a perpendicular from the upper left corner to the base. This marks a right triangle at the left end.

Cut off that triangle. Slide it to the right end, where it fills the sloping gap. The pieces now make a rectangle. Nothing was added, lost or overlapped.

The rectangle has the same base and height as the parallelogram. This gives:

Here, $b$ is the base length and $h$ is its perpendicular height. A parallelogram with base 9 cm and height 4 cm becomes a 9 cm by 4 cm rectangle:

Why choose a longer side for the cut? With a short base, a very strong lean can put the height beyond its end. There is then no end triangle to cut off at that point. Using a longer side makes the cut possible. The area formula itself works for every base with its own height.

Why the slanted side does not work

Give the parallelogram slanted sides of 5 cm. Its height is still 4 cm. Multiplying 9 by 5 would find a rectangle taller than the one made by rearranging:

That is too large. The leaning side measures along an edge. The height measures the gap between the parallel bases.

Same height, different lean

Hold a 7 cm base in place. Keep the opposite side 4 cm above it. Slide the upper side sideways. Every parallelogram you make has area 28 cm² because the base and height stay fixed.

The perimeter can change. A 7 cm by 4 cm rectangle has perimeter 22 cm. A parallelogram with the same base and height but 5 cm slanted sides has perimeter 24 cm. The surface can stay equal while the edges get longer.

A rhombus has four equal sides and is also a parallelogram. With a side of 5 cm and a matching height of 4 cm, its area is 20 cm². On a grid, a parallelogram six units across and three rows tall has area 18 square units. Its sloping end pieces can be joined to make whole columns of squares.

Once you have a matching base and height, the calculation is one multiplication. The main choice happens before that: deciding which measurements belong in it.

Area and perimeter need different lengths

A parallelogram has base 10 m, slanted sides 6 m and height 5 m. For area, use the base and its height:

For perimeter, add the four edges:

The 6 m side matters for the edge length. It does not replace the 5 m height in the area calculation.

Use either matching pair

Another parallelogram has sides 8 cm and 5 cm. The height to the 8 cm side is 4 cm:

The height to the 5 cm side is 6.4 cm:

Both pairs measure the same shape. Mixing the 8 cm base with the 6.4 cm height would combine lengths from different pairs.

Work with decimals and fractions

For a base of 7.5 m and height of 3.2 m:

For a base of 3½ ft and height of 2½ ft, change the mixed numbers to fractions:

The shape uses the same formula; the numbers need different arithmetic.

Match units before multiplying

A base is 1.2 m and its height is 40 cm. In meters, the height is 0.4 m:

In centimeters, the base is 120 cm:

These areas agree because one square meter contains 10,000 square centimeters.

Use the area to plan materials

A garden bed has base 12 ft and height 5 ft. Its area is 60 ft². One bag of mulch covers 12 ft², so the bed needs:

With slanted sides of 6 ft, the bed is less tall than those sides are long. Its area must be below the area of a 12 ft by 6 ft rectangle, or 72 ft². The result of 60 ft² fits this check.

Draw a diagonal across a rectangle. You get two identical right triangles. Each triangle covers half of the rectangle's surface.

Start with a right triangle

An 8 cm by 3 cm rectangle has area 24 cm². Its diagonal divides that surface equally, so each right triangle has area 12 cm². The two legs of each triangle are the rectangle's length and width.

Make a copy of any triangle

The same idea works beyond right triangles. Make an identical copy of a triangle. Turn the copy halfway around, through 180°. Join the two triangles along a matching side to make a parallelogram.

The parallelogram has the same base and height as the original triangle. Its area is base times height. Each triangle is half of that total:

In words: multiply a base by its matching height, then halve once.

For an acute triangle, draw its inside height. This makes two right triangles. A rectangle around the triangle splits into two smaller rectangles along the height. Each right triangle fills half of its small rectangle. Together they fill half of the large rectangle.

An obtuse triangle can have an outside height. Two copies still make a parallelogram with the same base and height, so the same formula works.

Halve once, in any order

For a triangle with base 10 units and height 7 units:

You can first multiply 10 by 7 and divide 70 by 2. Or halve 10 to get 5, then multiply 5 by 7. Both give 35.

Halving both lengths would give 5 times 3.5, or 17.5. That is a quarter of the original product. Only one halving belongs in the triangle formula. Leaving out the halving gives 70, the area of both copies together.

Move the opposite corner sideways

Keep a triangle's base 8 cm long. Move the opposite corner along a line 5 cm above the base. The height remains 5 cm even as the triangle changes shape. Every triangle formed this way has area 20 cm².

Its sides and perimeter may change. Its area stays fixed because the base and perpendicular height stay fixed.

A triangle's shape can help you find its height. The formula stays the same for every triangle:

Find an inside height

An acute triangle has base 12 cm and height 7 cm. The height runs from the opposite corner to the base at 90°:

On a grid, count the base along its row. Count the height across the rows between the base line and the opposite corner. A six-unit base and a four-row height give 12 square units. The corner can move sideways along its row without changing that area.

Use the legs of a right triangle

A right triangle has legs of 9 in and 12 in. Its longest side is 15 in. The two legs meet at 90°, so they make a base–height pair:

The height to the 15 in side is 7.2 in. Using that pair gives the same result. The 9 in leg does not meet the 15 in side at a right angle, so those two lengths cannot be paired.

Pairing the 15 in side with the 9 in leg would incorrectly give 67.5 in². Their meeting angle is not a right angle.

Measure an outside height

An obtuse triangle has base 5 m and height 8 m. The perpendicular lands on a dashed extension outside the triangle:

Use the 5 m side itself. The extension is a guide for measuring the height, not an extra piece of the base.

Apply the rule with other numbers

A base of 7½ in and height of 4 in give:

A base of 4.6 cm and height of 3 cm give 6.9 cm². For a triangular sail with base 3 m and height 5 m, the area is 7.5 m². That is the cloth covering the triangle; seams need extra cloth.

A triangle needs twice the base of a rectangle to match its area at the same height. A 5 by 6 rectangle covers 30 square units. A triangle with base 10 and height 6 covers the same area. A parallelogram with base 5 and height 6 also covers 30 square units.

A trapezoid is a four-sided shape with at least one pair of parallel sides. Those parallel sides are its bases, written $b_1$ and $b_2$. The other two sides are its legs. Its height is the perpendicular gap between the bases.

Some books define a trapezoid as having exactly one pair of parallel sides. The area formula works with either definition.

Double the trapezoid

Make an identical copy, turn it 180°, and place it beside the first trapezoid. The two shapes fit into a parallelogram. Its base combines a long base and a short base, so its length is $b_1+b_2$. Its height is still $h$.

The parallelogram's area is $(b_1+b_2)h$. One trapezoid is half:

Split it along a diagonal

Draw a line between opposite corners. The trapezoid becomes two triangles. One triangle uses the first base; the other uses the second. Both have height $h$ because the bases are parallel.

Add their areas:

This explains why both bases must be included, while the legs are left out.

Use a rectangle and two triangles

An isosceles trapezoid has bases 10 cm and 6 cm, height 4 cm, and equal legs. Draw heights from the ends of the shorter base. The middle is a 6 cm by 4 cm rectangle. The remaining width is 4 cm, shared equally by two end triangles. Each has base 2 cm and height 4 cm.

The rectangle's area is 24 cm². Each triangle's area is 4 cm². Altogether:

The formula gives half of 16 times 4, also 32 cm².

Think of the average base

The average of the bases is their sum divided by two. Here the average is 8 cm. A rectangle 8 cm long and 4 cm tall has the same area as the trapezoid.

This also connects the formulas. Equal bases give a parallelogram: the average base is just the base itself. If the shorter base shrinks to a point, the shape becomes a triangle. The formula then uses half of the remaining base times height.

Find the parallel sides before doing any arithmetic. A trapezoid can be turned sideways, so its bases may run up and down. Its height then runs across the page.

Add, multiply, halve

For bases 7 cm and 11 cm and height 5 cm, start inside the parentheses:

Multiply that sum by the height, then halve:

Keep the bases together. Halving 7 and then adding 11 times 5 would give 58.5, which changes the formula.

Pick the perpendicular measurement

A right trapezoid has bases 9 ft and 6 ft. One leg meets both bases at 90° and is 4 ft long. The slanted leg is 5 ft. Its height is 4 ft:

The slanted leg belongs in the perimeter, not in this area calculation.

Check between two areas

Multiply the shorter base by the height: 6 times 4 gives 24 ft². Multiply the longer base by the height: 9 times 4 gives 36 ft². The trapezoid uses the average base, so its area must lie between these results. The answer of 30 ft² does.

Fractions, decimals and perimeter

Bases 2.4 m and 3.6 m add to 6 m. With height 1.5 m:

Bases 4½ in and 7½ in add to 12 in. With height 3 in, the area is 18 in².

An isosceles trapezoid with bases 10 cm and 4 cm, legs 5 cm and height 4 cm has area 28 cm². Its perimeter is a separate calculation:

The height is inside the shape, so it is not another edge to add.

Plan how much to buy

A deck has parallel edges 20 ft and 14 ft, separated by 10 ft. Its area is:

One can of stain covers 50 ft². Dividing gives 3.4 cans. You need 4 whole cans: three cover only 150 ft² and leave part of the deck unstained.

A diagonal joins opposite corners of a four-sided shape. In kites and rhombuses, the diagonals cross at right angles. These lengths give a useful area formula.

Meet the shapes

A kite has two pairs of equal sides next to each other. The diagonal joining the corners where equal sides meet is a line of symmetry. It cuts the other diagonal in half.

A rhombus has four equal sides. It is a parallelogram, and its diagonals cut each other in half at right angles. A square is both a rhombus and a kite.

Add two triangle areas

Use one full diagonal as the base of two triangles, one on each side. Their heights are the two parts of the other diagonal. Adding those heights gives the second full diagonal.

Writing the diagonals as $d_1$ and $d_2$ gives:

You can also draw a rectangle around the kite, with sides as long as the diagonals. The diagonals divide it into four smaller rectangles. The kite fills half of each, so it fills half of the whole rectangle.

For diagonals 10 cm and 6 cm:

Use the full lengths, from corner to corner. Multiplying two half-diagonals would miss part of the shape.

Compare two methods for a rhombus

A rhombus has diagonals 8 cm and 6 cm. Its area is 24 cm². Its sides are 5 cm long, with perpendicular height 4.8 cm. Base times height gives:

A square with both diagonals 6 cm long has area 18 cm² by the diagonal rule.

Check what the diagonals tell you

The half-product rule needs perpendicular diagonals. A 6 cm by 8 cm rectangle has two diagonals of 10 cm, but they do not meet at 90°. Its area is 48 cm² from length times width. Half of 10 times 10 would wrongly give 50.

Kite side lengths alone do not fix its area. Two kites can have the same four side lengths but different widths. Use their diagonals to measure their surfaces.

A toy kite with perpendicular crossbars of 80 cm and 50 cm needs 2,000 cm² of fabric to cover its shape. That is 0.2 m² because one square meter holds 10,000 square centimeters.

A composite shape, also called a compound shape, combines simpler shapes. A house outline may be a rectangle with a triangle above it. Each part has an area rule you already know.

Draw a useful split

Sketch the outline and draw lines to make rectangles, triangles, parallelograms or trapezoids. Label each part's own base and height. The parts must fill the whole shape without gaps or overlaps.

A house-shaped surface has a rectangle 10 m wide and 6 m tall. Its triangular top has base 10 m and height 4 m.

1. Rectangle: $10\times 6=60{\text{ m}}^2$.
2. Triangle: $\frac{1}{2}\times 10\times 4=20{\text{ m}}^2$.
3. Whole surface: $60+20=80{\text{ m}}^2$.

The roof height starts at the top of the rectangle. Using the full height of the house as the triangle's height would count extra surface.

Find a part's missing length

A rectangle is 8 cm long and 5 cm tall. A right triangle joins its right end, sharing the full 5 cm vertical edge. The whole bottom edge measures 12 cm.

The triangle's bottom base is the part left after the rectangle:

The rectangle has area 40 cm². The triangle has area half of 4 times 5, or 10 cm². Their total is 50 cm². Using 12 cm as the triangle's base would count the rectangle's width again.

Choose another split

Several correct splits can measure the same surface. A trapezoid can be split along a diagonal into two triangles. An isosceles trapezoid can also be split into a rectangle and two right triangles. Every complete split gives the same total.

A gap leaves area uncounted. An overlap counts some area twice. Trace each part's edge on your sketch to check that every region appears exactly once.

Split polygons into triangles

A polygon is a closed shape with straight sides. It can be split using diagonals that stay inside and do not cross. In a convex polygon, every corner points outward. Drawing diagonals from one corner to all nonneighboring corners makes triangles.

A concave polygon has an inward corner. Some diagonals from one corner may leave the shape. Choose inside diagonals that make a complete split.

A regular hexagon splits from its center into six identical triangles. With sides 6 cm and center-to-side distance about 5.2 cm, each triangle has area about 15.6 cm². Six together have area about 93.6 cm². The answer is approximate because the height was rounded.

A hole or notch can make adding awkward. It may be easier to measure a larger shape first. Subtract the surface that does not belong to the region you need.

Remove a cutout

A rectangular lawn measures 20 ft by 12 ft. A triangular flower bed, with base 6 ft and height 4 ft, replaces part of the grass.

First find the full rectangle:

The flower bed covers:

The grass covers $240-12=228{\text{ ft}}^2$. Subtract areas, not the lengths around their edges.

A frame works the same way. A photo is 30 cm by 20 cm. The frame's outside measures 40 cm by 30 cm. The outer area is 1,200 cm² and the photo area is 600 cm². The frame alone covers 600 cm². The outer rectangle already includes the photo, so adding the two areas would count it twice.

Enclose a tilted triangle on a grid

A triangle has corners at (0, 0), (6, 2) and (2, 5). The first number counts across; the second counts up. None of its sides follows a grid line, so its perpendicular height is awkward to read.

Draw a rectangle from (0, 0) to (6, 5). It contains 30 square units. Three right triangles sit outside the tilted triangle but inside this rectangle:

• Below the edge from (0, 0) to (6, 2): base 6, height 2, area 6.
• At the upper right: base 4, height 3, area 6.
• At the upper left: base 2, height 5, area 5.

They fill all the space around the triangle. Subtract their total:

Choose the shorter route

A right trapezoid with bases 15 ft and 9 ft and height 8 ft fits in a 15 ft by 8 ft rectangle. The missing corner is a right triangle with legs 6 ft and 8 ft. Subtracting gives $120-24=96{\text{ ft}}^2$.

Adding works well for pieces sitting beside each other. Subtracting works well for holes, notches and cut corners. Choose the method that uses lengths you can find. The remaining area must be smaller than its outer area. With several separate cutouts, subtract each one once.

An area answer may be partway through a problem. To buy paint or flooring, you also need the coverage or price. Start by deciding which surface needs the material.

Paint a wall with a door

A shed's end wall is a rectangle 8 ft wide and 7 ft tall. A triangular top adds another 3 ft of height. A 3 ft by 6 ft door will not be painted.

Find the two wall parts:

Subtract the door's area:

One can covers 20 ft². The number of cans is $50÷20=2.5$. Buy 3 whole cans. Two cover only 40 ft²; three cover enough.

The wall's peak is 10 ft above the ground. The whole wall fits inside an 8 ft by 10 ft rectangle. Its area to paint must be below 80 ft². This check would quickly reject a result of 500 ft².

Turn an area into a price

A trapezoid lawn has parallel sides 30 ft and 40 ft, separated by 25 ft. Its area is:

New grass costs $0.40 per square foot. Multiply the number of square feet by that price:

The grass costs $350. The square-foot unit in the price matches the square-foot area.

Match the area unit to the sale unit

A room measures 12 ft by 15 ft, so its area is 180 ft². Carpet is priced by the square yard. One square yard contains nine square feet:

Use 20 with a square-yard price. Using 180 with that price would charge for nine times the actual surface.

Decide how to round

Whole cans, bags and rolls need rounding up when a fraction remains. An estimate may instead ask for the nearest sensible value. Keep the area calculation accurate until the final rounding step.

Finish with the quantity the question asks for: “You need 3 cans of paint,” or “The grass costs $350.” A number of square feet alone does not answer a question about cans or cost.

An area formula links lengths to a surface area. When one length is missing, undo the operations that used it. The missing value is a length, so its unit is centimeters or feet, not square units.

Undo multiplication in a parallelogram

A parallelogram has area 56 cm² and base 8 cm. Since base times height gives area, divide by the base:

Check by finding the area again: $8\times 7=56{\text{ cm}}^2$. To find a missing base, divide the area by the known height instead.

Double a triangle's area first

A triangle has area 30 m² and base 12 m. Its area is half of base times height. Double the area to undo that halving:

Now divide by the known base:

Check: half of 12 times 5 is 30 m².

Dividing 30 by 12 straightaway gives 2.5 m. That height would produce only 15 m², half of the required area. The doubling step is essential for triangles.

You can write the same work as a simple equation:

So $h=30÷6=5$. Here, halving the known base first gives the number to divide by.

Find a second height of the same shape

A parallelogram has sides 10 cm and 6 cm. The height to the 10 cm side is 3 cm. Its area is 30 cm². Using the 6 cm side as the new base:

The new base needs its own height. You cannot keep the first height after changing bases.

A right triangle with legs 15 cm and 20 cm has area 150 cm². Its longest side is 25 cm. The height to that side is:

Missing bases and fractional areas

The steps also find a triangle's base. A triangle has area 10½ in² and height 3 in. Double the area to get 21, then divide by 3. The base is 7 in. Checking half of 7 times 3 gives the required 10½ in².

Working backward can take more than one step. First decide what the formula gives you: a height, a whole diagonal, or the sum of two bases.

Recover a trapezoid height

A trapezoid has area 52 cm² and bases 5 cm and 8 cm. Their sum is 13 cm. Double the area, then divide by this sum:

For any trapezoid, $h=2A÷(b_1+b_2)$.

Find a missing base

Another trapezoid has area 60 cm², height 8 cm and one base 6 cm long. Double the area and divide by the height:

This is both bases together. Subtract the known one:

Check: half of 15 times 8 gives 60 cm². Forgetting to double would give a base of 1.5 cm and an area of only 30 cm².

A base length must be positive. A zero or negative result cannot make the stated trapezoid. Check the arithmetic and the measurements.

Recover a full diagonal

A kite has area 40 m² and one diagonal 10 m long. Since its area is half the product of the diagonals:

This is the full distance between the other pair of opposite corners. Checking half of 10 times 8 gives 40 m². The same method works for a rhombus.

Separate the unknown part

A house shape has total area 126 m². Its rectangular part is 14 m wide and 7 m tall. The triangle on top therefore covers:

The triangle's base is 14 m, so its height is $2\times 28÷14=4\text{ m}$. Using the whole 126 m² in the triangle formula would treat the rectangle as roof area too.

Design shapes with one area

A triangle of area 24 cm² needs base times height to equal 48. Bases and heights of 6 and 8, 12 and 4, or 16 and 3 all work. A parallelogram of area 24 cm² instead needs a product of 24.

Area alone does not choose a single shape. Another measurement can narrow the design to a particular base–height pair.