Volume of Cubes and Rectangular Prisms

A brick takes up space on your desk. So does a pillow. Volume tells you how much space an object takes up. For a hollow box, you can also measure the space inside.

Length, area and volume

These measurements answer different questions. Think about a shoebox.

A box has length, width and height. Its volume depends on all three directions. A tall, narrow box can hold less than a low, wide one.

Compare by filling

You can compare what two containers hold using dry rice. Fill container A to the top. Pour all that rice into empty container B.

• If rice spills over B, A holds more.
• If B still has empty space, B holds more.
• If B fills exactly, both hold the same amount.

The rice lets you compare the inside spaces even when the containers look different.

Move pieces without changing volume

Build a tower with 12 matching cubes. Now rebuild those same cubes as a flat wall. You changed the shape, but you still have 12 cubes. The volume stays the same. Squashing a ball of clay also keeps its volume when you add or remove no clay.

Height alone can fool you. A tower 1 cube wide, 1 cube deep and 10 cubes tall uses 10 cubes. A shorter block, 3 cubes long, 2 cubes wide and 2 cubes tall, uses:

The shorter block has more volume because it spreads out farther.

Name the box shapes

A rectangular prism has six rectangular faces. Opposite faces match. It has 12 edges and 8 corners, called vertices. A shoebox is a familiar example. You may also hear the British name, cuboid.

A cube is a special rectangular prism. Its six faces are squares, and all its edges match in length.

Volume and weight tell you different things. A big pillow can take up more space than a brick. The brick can still be heavier.

You can measure volume by filling a shape with cubes. The cubes must all match in size. Then each one counts as the same amount of space.

Meet the unit cube

A unit cube has edges that are each 1 unit long. Its volume is 1 cubic unit. If each edge is 1 centimeter long, its volume is 1 cubic centimeter, written 1 cm³.

Pack 18 centimeter cubes into a box with no empty spaces. If they fill it exactly, the inside volume is 18 cm³.

Leave no gaps or overlaps

Fit cubes face to face. A gap is space you have not counted. An overlap would count some space twice.

Marbles leave spaces between them, even when they touch. Counting marbles tells you how many marbles fit. It does not measure all the space inside the box exactly.

Count whole cubes, including hidden ones

A drawing only shows the outside of a model. In a solid block, cubes also fill the back, bottom and middle. You can build the model or imagine removing one layer at a time.

Be careful with the little squares on its faces. One cube can show several faces in a drawing. Those faces still belong to just one cube.

For example, a solid block 2 cubes long, 2 cubes wide and 2 cubes tall has 8 cubes. A drawing can show three faces with 4 squares each. Counting those 12 squares would count faces, not the 8 cubes.

A staircase is different from a full box. If its bottom row has 3 cubes, its next row has 2, and its top row has 1, add the rows:

The staircase has a volume of 6 cubic units. Do not fill its empty steps in your head.

Keep the measuring unit the same

The same box needs more small cubes than large cubes. A cube with edges twice as long takes the place of 8 smaller cubes:

So a space filled by 24 small cubes can hold 3 of those large cubes. The space has not shrunk. Only the measuring pieces changed. When you compare cube counts, use cubes of the same size.

Counting cubes one at a time can take a while. A rectangular prism gives you a shortcut: every full layer has the same number of cubes.

Start with one layer

Picture a box with 5 cubes along its length. There are 3 rows across its width. Each row contains 5 cubes, so one layer contains:

This is like finding the area of a rectangle. The bottom covers 15 square units. One cube stands on each square, so that layer contains 15 cubes.

Stack the layers

The box is 4 cubes tall. That means it has 4 equal layers. Count 15, 30, 45, 60, or multiply:

The volume is 60 cubic units. The first multiplication counted one layer. The second counted how many times that layer repeats.

Turn the box

You can count the same block from another side. Stand it so the layers are 3 cubes by 4 cubes. Now each layer has 12 cubes, and there are 5 layers:

Turning a block changes your view, not the cubes inside it.

Use clues from a partly filled box

Suppose 6 cubes fit along the length of a box. Two rows fit across it, and 3 layers fit up the side. You do not need to fill every position before finding its full volume.

One layer holds:

Three layers hold:

Adding 6, 2 and 3 would only add edge counts. It would miss almost all the cubes inside.

Make an estimate

An estimate uses close, easy numbers when you cannot count exactly. About 10 sugar cubes fit along a box. You judge that about 6 rows and 4 layers fit.

The box holds about 240 sugar cubes. Because the row and layer counts were estimates, say “about” with your answer.

You do not need to draw every cube inside a rectangular prism. Measure the three edges that meet at one corner. Call them length, width and height.

Write the volume formula

Use:

Here, $V$ means volume, $l$ length, $w$ width and $h$ height. With whole-number edge lengths, length times width counts one layer of unit cubes. Height counts the layers.

A box is 8 cm long, 5 cm wide and 3 cm high. First find one layer:

Then count all 3 layers:

The volume is 120 cm³. Multiplying only 8 and 5 gives the area of the bottom. Adding all three lengths gives 16 cm, a length total. Neither gives the box’s volume.

Choose an easy order

You can multiply the three numbers in any order. For a box with edges 5 cm, 7 cm and 4 cm, pair 5 and 4 first:

The volume is 140 cm³. Turning this box on its side changes which edge you call the height. It does not change the product.

Use equal edges for a cube

All edges of a cube have the same length, called $s$. Its formula is:

A cube with 4 cm edges has volume:

That is 64 cm³. The small 3 means “use the edge length as a factor three times.” It does not mean multiply the edge length by 3.

Match the length units first

A box measures 1 ft by 6 in by 4 in. Change 1 ft to 12 in before multiplying:

The volume is 288 in³. Using 1, 6 and 4 together would mix feet and inches.

What about a decimal edge?

A box measures 2.5 cm by 4 cm by 3 cm:

Its volume is 30 cm³. Whole centimeter cubes cannot fit the 2.5 cm edge exactly. But half-centimeter cubes can fill it: 5 along, 8 across and 6 up make 240 small cubes. Eight small cubes fill 1 cm³, so 240 small cubes fill 30 cm³. Smaller cubes explain why the formula still works.

Sometimes a problem tells you the area of the bottom face. You can use that area straight away. You do not have to find the length and width separately.

A second form of the formula

Call the area of the base $B$. Then:

For a rectangular base, $B=l\times w$. So this is the same three-edge formula, with the first multiplication already done.

With whole-number edges, picture a cube standing on each unit square of the base. A base of 24 cm² has room for 24 centimeter cubes in one layer. A height of 5 cm gives 5 such layers:

The box’s volume is 120 cm³. The base-area formula works for decimal measurements, too.

Pick a face and its matching height

Any face of a rectangular prism can be the base. The height must go straight out from that face, at a right angle. It is the third edge, not an edge you already used in the base area.

Take a box 8 cm by 5 cm by 3 cm. Its possible bases give these pairs:

For the 8 cm by 5 cm base, use 3 cm as the height. Using 5 cm again would count one direction twice and leave out another.

Follow the units

Base area uses square units. Height uses length units. Their product uses cubic units. A 24 cm² base multiplied by a 5 cm height gives 120 cm³, not 120 cm².

The same method works with inches. A fish tank floor has an area of 600 in². The water is 10 in deep:

The water’s volume is 6,000 in³. Here the height is the water depth, because you are measuring the water inside the tank.

A volume number needs a unit. “20” alone does not tell you whether you mean a tiny space or a large one. Picture the cube named by the unit.

Build a picture of the unit

A cubic centimeter, or cm³, is the volume of a cube with 1 cm edges. The little ones cube in most base-ten block sets is this size. Small boxes can be measured in cm³.

A cubic meter, or m³, is the volume of a cube with 1 m edges. Picture a large box, 1 meter long, wide and tall. Use m³ for large spaces, such as rooms or swimming pools.

The raised 3 tells you that the unit measures space in three directions. Read cm³ as “cubic centimeters” and m³ as “cubic meters.” Do not read the raised 3 as “times 3.”

Choose among US customary units

A cubic inch, written in³, comes from a cube with 1-inch edges. A cubic foot, ft³, comes from a cube with 1-foot edges. A cubic yard, yd³, comes from a cube with 1-yard edges.

Use in³ for a small box. Refrigerator space is often given in ft³. Garden soil and mulch are often sold in yd³.

Very small and very large spaces

A cubic millimeter, mm³, comes from a cube with 1 mm edges. It can describe very tiny volumes. A cubic kilometer, km³, comes from a cube with 1 km edges. It can describe huge volumes, such as water in a large lake.

Choose a unit that makes the size easy to picture. Measuring a room in cm³ is valid, but it gives a very large number. Cubic meters are easier to work with.

Match the kind of measurement

A box’s volume cannot be 120 cm². That symbol describes flat area.

Estimate with a familiar cube

A classroom about 10 m long, 8 m wide and 3 m high has volume:

That is about 240 m³. Imagine 240 cubes with 1 m edges filling the room. This picture helps you judge whether a volume answer makes sense.

A box keeps the same volume when you change its unit. The number changes because you are counting a different size of cube.

Convert all three directions

You know that 1 m is 100 cm. A cubic meter has three edges to change. In centimeters, the unit cube is 100 cm long, 100 cm wide and 100 cm tall:

So 1 m³ = 1,000,000 cm³. Using just 100 converts only one direction. Using 10,000 converts two directions, which is the factor for square units. Volume needs all three.

Find other cubic conversion factors

Each row starts with one large cube. Count how many smaller cubes fill it.

Decide whether to multiply or divide

When you change to smaller cubes, you need more of them. Multiply by the cubic conversion factor:

So 3 m³ is 3,000,000 cm³. Also, 2 yd³ is 54 ft³, because each cubic yard contains 27 cubic feet.

When you change to bigger cubes, divide:

So 3,456 in³ is 2 ft³. In the same way, 5,000,000 cm³ is 5 m³.

Choose a convenient route

For a box 3 ft by 2 ft by 1 ft, multiplying the edges gives 6 ft³. Convert this volume by multiplying 6 by 1,728 to get 10,368 in³.

You can also change the edges first: 36 in, 24 in and 12 in. Then:

Both routes describe the same box. If your conversion to a smaller unit gives a smaller number, check whether you divided by mistake.

A container has walls as well as an empty space inside. Capacity means how much the container can hold. It is the volume of that inside space.

Link cubic centimeters and milliliters

One milliliter takes up exactly one cubic centimeter:

So an inside volume of 250 cm³ holds 250 mL. Both units describe an amount of space. You can use liters for rice or sand, and cubic meters for water.

A liter is 1,000 mL. Picture a cube with inside edges of 10 cm:

Its inside volume is 1,000 cm³, so its capacity is exactly 1 L. A large water bottle may hold 1 L. A teaspoon holds about 5 mL.

A cubic meter is much larger. It contains 1,000,000 cm³, or 1,000,000 mL. Dividing by 1,000 gives 1,000 L.

Measure the inside

A fish tank measures 50 cm by 30 cm by 40 cm on the inside. Its capacity is:

It holds 60,000 cm³, which is 60,000 mL. Change milliliters to liters:

The tank holds 60 L when full to that height.

Outside measurements include the walls. A closed hollow cube is 12 cm across outside, with walls 1 cm thick on opposite sides. Its inside edge is 10 cm. Its inside space holds 1 L, even though its outside box measures 1,728 cm³. You cannot fill the walls with water.

Measure a stone using water

A measuring jug can help you find an uneven object’s volume. Choose an object that sinks completely and does not soak up water. Read the water volume before and after putting it fully underwater. It pushes aside its own volume of water.

If the jug reading changes from 400 mL to 550 mL:

The stone’s volume is 150 cm³. The final reading includes the water already in the jug. Subtract the starting reading to find the extra space taken up by the stone.

Sometimes you know a box’s volume but not all its edges. Think about how many cubes fit in each layer. Then find how many layers you need.

Find a missing height

A box has volume 72 cm³. It is 4 cm long and 3 cm wide. One layer of centimeter cubes contains:

Divide the total cubes by the cubes in each layer:

The height is 6 cm. Check by multiplying all three edges:

For any missing edge, divide the volume by the product of the other two edges. If the base area is already known, use:

A room with volume 180 m³ and floor area 30 m² is 6 m high.

Find a cube’s edge

A cube with volume 125 cm³ needs an edge length used as a factor three times. Try nearby whole numbers:

Its edge is 5 cm. Dividing 125 by 3 would not undo cubing. You need three equal factors whose product is 125.

Same volume, different shape

Boxes can hold the same number of unit cubes and still look different. These are all the different whole-number box sizes with volume 24 cm³, counting turned versions as the same box:

To compare two boxes, calculate both volumes. A 6 cm by 5 cm by 4 cm box holds 120 cm³. An 8 cm by 4 cm by 4 cm box holds 128 cm³. Being wider does not make the first box larger in volume.

Change one edge or all three

Start with a 2 cm by 3 cm by 4 cm box. It holds 24 cm³. Double only its length:

Its volume doubles. Double all three original edges instead:

Now the volume is 8 times as large. There are twice as many cubes along each of three directions, giving a factor of $2\times 2\times 2=8$.

A stepped solid may not be one rectangular prism. You can still measure its volume using smaller rectangular prisms. Draw a dividing line so the pieces fit together without overlapping.

Label a stepped block

Picture an L-shaped block that is 4 cm deep. From the front, it is 10 cm wide. The bottom step is 3 cm high across the full width. On the left, a 4 cm wide tower reaches a total height of 8 cm.

Split the shape along the top of the bottom step. The tower above that step is:

It is 5 cm high, not 8 cm. The first 3 cm of height already belong to the bottom piece.

Find each piece, then add

The bottom piece measures 10 cm by 3 cm by 4 cm:

The upper tower measures 4 cm by 5 cm by 4 cm:

Add the two volumes:

The whole block has volume 200 cm³.

Try a different cut

You could use a full-height left column instead. It measures 4 cm by 8 cm by 4 cm, giving 128 cm³. The lower right piece has width:

Its volume is $6\times 3\times 4=72$ cm³. Adding 128 and 72 again gives 200 cm³. The cuts differ, but both count every part once.

Subtract an empty corner

Another route is to picture the full box around the block. That box measures 10 cm by 8 cm by 4 cm, so its volume is 320 cm³.

The empty upper-right corner measures 6 cm by 5 cm by 4 cm. Its volume is 120 cm³. Subtract it:

Avoid counting the corner twice

If you add the whole bottom piece and the full-height left column, they overlap. Their shared corner measures 4 cm by 3 cm by 4 cm, or 48 cm³.

The mistaken total of 248 cm³ includes that corner twice. Take away one copy of 48 cm³ to get 200 cm³. It is easier to draw non-overlapping pieces before calculating.

Start with what you need to find. Is it a volume, a capacity, a length or a number of cubes? Sketch the box and label its measurements. Make the length units match before using a formula.

Fill to the right depth

A garden bed is 8 ft long and 3 ft wide. The soil should be 1 ft deep:

You need 24 ft³ of soil. Use the depth of the material you are measuring.

A tank has inside measurements 60 cm by 40 cm by 50 cm. Its water is only 30 cm deep. The water volume is:

That is 72,000 cm³, or 72 L. Using 50 cm would find the full tank’s capacity, 120 L.

Find a rise in water depth

The tank’s base area is 2,400 cm². Pour in 12 L more water. First change that to 12,000 cm³. Spread that extra volume over the base:

The water rises 5 cm, from 30 cm to 35 cm. There is still space below the tank’s 50 cm top.

For a tank 1 m long, 50 cm wide and 40 cm tall, first change 1 m to 100 cm. Its full capacity is 200,000 cm³, or 200 L.

Pack whole cubes

How many 2 cm cubes fit in a box measuring 6 cm by 4 cm by 5 cm, in straight rows with cube edges along box edges?

• Along the length, 3 cubes fit.
• Across the width, 2 cubes fit.
• Up the height, only 2 full cubes fit. There is 1 cm left.

You can pack 12 cubes. Dividing the box volume, 120 cm³, by one cube’s volume, 8 cm³, gives 15. But those extra cubes cannot fit into the 1 cm gap at the top.

When every edge fits a whole number of cube edges, division of volumes works too. A 6 cm by 4 cm by 6 cm box holds 18 of the 2 cm cubes and has no leftover space.

Check the size of your answer

For a 19 cm by 11 cm by 9 cm box, estimate using 20, 10 and 9. That gives about 1,800 cm³. The exact answer, 1,881 cm³, is close. Finish with the requested unit: a tank answer in liters needs one more conversion after finding cm³.