Volume Is Additive: Split a Solid into Two Prisms | Grade 5 Math | LevelKite

You already know how to find the volume of a right rectangular prism, a box shape whose six faces are all rectangles. You multiply its length, width and height.

Many solids are not one box. Think of two steps of a staircase, or a block shaped like the letter L. A composite solid is a solid made of two or more simpler solids joined together. In this lesson, each composite solid is made of two right rectangular prisms.

You can find the volume of a composite solid with what you already know, plus one new idea about how volume works. Start with the solid in the figure.

Bottom layer: 6 cubes. Top layer: 4 cubes.

6 + 4 = 10

cubic units.

The solid in the figure is a stack of unit cubes. A unit cube has edges $1$ unit long and fills $1$ cubic unit of space. So the volume of the solid is the number of cubes in it.

Count the cubes in two parts:

The bottom layer is one part. It has $6$ cubes.
The top layer is the other part. It has $4$ cubes.

The two parts do not overlap. That means no cube belongs to both parts. Together the two parts make the whole solid, so you miss no cube. You count every cube exactly once, and the volume is $6 + 4 = 10$ cubic units.

If the two parts shared a cube, you would count that cube twice, and the sum would be too big.

Key concept

Volume is additive

When a solid is made of two parts that do not overlap, the volume of the solid is the sum of the volumes of the parts. Additive means just that: you add the parts to get the whole. This works because volume counts the cubic units that fill the solid, and each cubic unit belongs to exactly one part.

\text{volume of the solid} = \text{volume of part 1} + \text{volume of part 2}

To use this idea, you need two parts whose volumes you can find. So you split the composite solid into two right rectangular prisms. To split a solid, picture one straight cut that divides it into two parts. Choose the cut so that each part is a right rectangular prism. Often more than one cut works.

Next, find the edge lengths of each prism: its own length, width and height. Take care with this step. A prism that is only part of the solid can have a shorter edge than the whole solid has. Read the length, width and height from the part, not from the whole solid.

The next figure shows a composite solid made of unit cubes. With cubes, you can find an edge length by counting the cubes along that edge.

Bottom layer: 6 cubes. Middle layer: 6 cubes. Top layer: 2 cubes.

6 + 6 + 2 = 14

cubic units.

Worked example

Split with a level cut

The solid in the figure is $3$ units long and $2$ units wide. One end is $3$ units high, and the rest is $2$ units high. Split the solid into two right rectangular prisms and find its volume.

Picture one flat, level cut just under the top layer. The bottom two layers form one prism. The top layer forms a second prism. No cube is in both prisms, and together the two prisms make the whole solid.
Find the edge lengths of the bottom prism by counting cubes along its edges. It is 3 units long, 2 units wide and 2 units high. Multiply the edge lengths to get its volume. The result matches the bottom two layers, which have 6 cubes each.
$3 \times 2 \times 2 = 12$
Find the edge lengths of the top prism. It sits on one end, so it is only 1 unit long, not 3. It is 2 units wide and 1 unit high. The result matches the 2 cubes in the top layer.
$1 \times 2 \times 1 = 2$
Add the two volumes, because the volume of the solid is the sum of the volumes of its two parts.
$12 + 2 = 14$

Answer

The volume of the solid is $14$ cubic units. That matches the $14$ cubes you can count in the figure.

Worked example

Split the same solid a different way

Use the same solid as in the last example. This time, split it with a straight up-and-down cut. Is the volume still $14$ cubic units?

Picture one straight up-and-down cut at the place where the height changes. The tall end forms one prism. The low part forms a second prism. Again, no cube is in both prisms, and together they make the whole solid.
Find the edge lengths of the tall prism. It is 1 unit long, 2 units wide and 3 units high. Multiply to get its volume. The result matches the figure: the tall prism has 2 cubes in each of the 3 layers.
$1 \times 2 \times 3 = 6$
Find the edge lengths of the low prism. The whole solid is 3 units long, and the tall prism uses 1 unit of that length, so the low prism is 2 units long. It is 2 units wide and 2 units high. In the figure, it has 4 cubes in each of the bottom two layers.
$(3 - 1) \times 2 \times 2 = 2 \times 2 \times 2 = 8$
Add the two volumes, because the two prisms do not overlap.
$6 + 8 = 14$

Answer

Yes. The volume is still $14$ cubic units. The parts changed from $12$ and $2$ to $6$ and $8$ , but the total stayed the same because the solid itself did not change.

Many solids come with measurements in a unit such as centimeters (cm) instead of cubes. Then you cannot count cubes, but the plan stays the same: split, find the edge lengths of each prism, multiply, and add.

Sometimes the problem does not give an edge length you need. You can find it from the lengths you know. You already did this with cubes: the whole solid was $3$ units long and the tall prism used $1$ unit, so the low prism was $3 - 1 = 2$ units long. With measurements, you subtract in the same way.

Worked example

A solid with measurements

A solid has the same step shape as the cube solid from the first two examples: one tall end and one low end. The whole solid is $10$ cm long and $4$ cm wide. The tall end takes up $3$ cm of the length and is $6$ cm high. The low end is $2$ cm high. What is the volume of the solid?

Split the solid with one straight up-and-down cut at the place where the height changes. You get a tall prism and a low prism that do not overlap.
Find the edge lengths of the tall prism. It is 3 cm long, 4 cm wide and 6 cm high. Multiply them to get its volume in cubic centimeters.
$3 \times 4 \times 6 = 72$
Find the edge lengths of the low prism. The problem does not give its length. The whole solid is 10 cm long, and the tall prism takes up 3 cm of that, so the low prism is 7 cm long. It is 4 cm wide and 2 cm high.
$(10 - 3) \times 4 \times 2 = 7 \times 4 \times 2 = 56$
Add the two volumes, because volume is additive.
$72 + 56 = 128$
Check with a different split. Picture a flat, level cut 2 cm above the bottom. The bottom prism is 10 cm long, 4 cm wide and 2 cm high. The top prism is 3 cm long, 4 cm wide and 4 cm high, because 6 cm take away 2 cm leaves 4 cm. Add the two volumes. The total is the same.
$10 \times 4 \times 2 + 3 \times 4 \times 4 = 80 + 48 = 128$

Answer

The volume of the solid is $128$ cubic centimeters. Both splits give the same total.

Takeaway: Volume is additive. To find the volume of a composite solid, split it into two right rectangular prisms that do not overlap. Find the edge lengths of each prism, multiply to get each volume, and add the two volumes. A different split gives the same total, because the solid fills the same space however you cut it. You can use a second split to check your work.

Next, you will use this idea to solve word problems about solids made of two prisms, compare volumes, and find and correct mistakes.