Volume of Prisms: Base Area Times Height | Grade 7 Math | LevelKite

You already know how to find the volume of a rectangular prism by counting unit cubes. Volume is the amount of space a solid fills. You measure it by the number of unit cubes, which are cubes with edges 1 unit long, that fill the solid with no gaps and no overlaps. That is why you write volume in cubic units, such as cubic centimeters ( $\text{cm}^3$ ).

In this lesson you will turn that counting into one rule. Then you will use the rule on cubes, rectangular prisms, triangular prisms and other right prisms.

Recall that a prism is a solid with two parallel faces of the same shape and size, called its bases. Each base is a flat shape with straight sides, such as a rectangle or a triangle. In a right prism, the side faces that join the bases are rectangles, and they meet the bases at right angles.

The height of the prism is the perpendicular distance between the two bases. Perpendicular means that you measure it at a right angle to both bases. Every prism in this lesson is a right prism.

Here is why counting cubes turns into multiplying. Take a rectangular prism whose base is a rectangle 5 cm long and 3 cm wide. Its height, perpendicular to the base, is 4 cm. Fill it with unit cubes that have 1 cm edges.

One layer. The base has an area of $5 \times 3 = 15$ square centimeters. One cube sits on each square centimeter, so the bottom layer holds $15$ cubes.
Number of layers. Each layer is 1 cm tall, so a height of 4 cm takes $4$ layers.
Equal layers. Slice the prism parallel to its base. The flat face made by the cut is called a cross-section, and it has the same shape and size as the base. So every layer holds $15$ cubes.

The prism holds $15 \times 4 = 60$ unit cubes, so its volume is $60 \text{ cm}^3$ . The number of cubes in one layer equals the base area in square centimeters. The number of layers equals the height in centimeters. You may know this as length times width times height: the first two lengths give the base area.

Key concept

Volume of a right prism

Let $B$ be the area of one base, in square units. Let $h$ be the height of the prism, the perpendicular distance between its bases, in the matching length unit. Let $V$ be the volume, in cubic units.

For any right prism, the volume is the base area times the prism height. $B$ is the number of unit cubes in a layer 1 unit thick, and $h$ is the number of layers. Parts of cubes and parts of layers count too.

The rule needs every length to be a positive number, all lengths to be in the same unit, and $h$ to be perpendicular to the base. A length that is not perpendicular to the base, such as a slanted edge, is not the height.

V = B \times h

Worked example

A rectangular prism

A right rectangular prism has a base that is a rectangle 8 m long and 4.5 m wide. The height of the prism, perpendicular to the base, is 3 m. Find the volume of the prism.

Name the base and check the measures. The base is a rectangle, the height is perpendicular to it, and all three lengths are positive and in meters. So the rule applies, and the volume will be in cubic meters.
Find the base area. A rectangle's area is its length times its width, because that counts the unit squares that cover it. The 4 whole meters of width give 32 squares. The extra half meter gives 8 half squares, which make 4 more.
$B = 8 \times 4.5 = 32 + 4 = 36 \text{ m}^2$
Multiply the base area by the prism height. A layer 1 m thick holds 36 cubic meters, and a height of 3 m makes 3 equal layers.
$V = B \times h = 36 \times 3 = 108 \text{ m}^3$
Check with a different base. Every face of a rectangular prism is a rectangle, so the 8 m by 3 m face can be the base, with 4.5 m as the height perpendicular to it. Multiplying the same three lengths in a different order gives the same product.
$(8 \times 3) \times 4.5 = 24 \times 4.5 = 108 \text{ m}^3$

Answer

The volume of the prism is $108 \text{ m}^3$ , which is $108$ cubic meters.

A cube is a rectangular prism whose six faces are all squares. A square is a rectangle with four equal sides, so a cube is not a new case. It follows the same rule.

Let $s$ be the edge length of a cube, a positive number in a length unit. Any face can be the base. Its area is $s \times s$ square units, and the height perpendicular to it is also $s$ . So the volume is $s \times s \times s$ cubic units, also written $s^3$ .

Take a cube with edges 6 cm long. The base is a square with area $6 \times 6 = 36 \text{ cm}^2$ , and the prism height is 6 cm. The volume is $36 \times 6 = 216 \text{ cm}^3$ . That is 6 layers of 36 unit cubes.

The layer idea does not need a rectangular base. In any right prism, every cross-section parallel to the base has the same shape and size as the base, so every layer 1 unit thick holds the same amount.

A prism whose bases are triangles is a triangular prism. Unit cubes do not fit neatly against every side of a triangle, so you have to cut some cubes. The pieces still add up. Above a whole square unit of the base there is a whole unit cube. Above half a square unit there is half a unit cube. So above a base with an area of $B$ square units, a layer 1 unit thick holds $B$ cubic units.

You can test this with numbers. A rectangular prism has a base 6 cm by 4 cm and a height of 10 cm, so its volume is $6 \times 4 \times 10 = 240 \text{ cm}^3$ . Stand it on its base and cut straight down along a diagonal of the base, from one corner to the opposite corner. You get two matching triangular prisms, so each has a volume of $240 \div 2 = 120 \text{ cm}^3$ .

Now use the rule on one of them. Its base is a right triangle whose perpendicular sides are 6 cm and 4 cm, so the base area is $\frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$ . The prism height is still 10 cm, and $12 \times 10 = 120$ . Base area times height gives $120 \text{ cm}^3$ , the same volume as half of the box.

Finding the base area: use positive lengths, all in the same unit

Base shape	Measures you need	Base area $B$ , in square units
Rectangle (a square is a rectangle too)	length and width	$\text{length} \times \text{width}$
Triangle	one side, and the triangle's height perpendicular to that side	$\frac{1}{2} \times \text{side} \times \text{height to that side}$
Trapezoid: four sides with at least one pair of parallel sides, so parallelograms count	the two parallel sides, and the perpendicular distance between them	$\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{distance between them}$
Another shape with straight sides, such as one with five sides	its area, when the problem gives it	the given area

Worked example

A triangular prism resting on its side

A right triangular prism rests on one of its rectangular faces. Its two bases are triangles with sides of 12 cm, 10 cm and 10 cm. The triangle's height, perpendicular to the 12 cm side, is 8 cm. The perpendicular distance between the two triangular bases is 15 cm. Find the volume of the prism.

Name the bases and the prism height. The bases are the two triangles, even though the prism rests on a rectangle. The prism height is the perpendicular distance between them, 15 cm. All the lengths are positive and in centimeters, so the volume will be in cubic centimeters.
Choose the measures for the base area. Use the 12 cm side with the 8 cm height, because that height is perpendicular to that side. The 10 cm sides are not perpendicular to the 12 cm side, so neither one is the height that goes with it.
Find the base area. A triangle has half the area of a rectangle with the same side and the same perpendicular height, so multiply 12 by 8 and take half.
$B = \frac{1}{2} \times 12 \times 8 = \frac{1}{2} \times 96 = 48 \text{ cm}^2$
Multiply the base area by the prism height. A layer 1 cm thick holds 48 cubic centimeters, and a height of 15 cm makes 15 equal layers. Split 15 into 10 and 5 to multiply.
$V = B \times h = 48 \times 15 = 480 + 240 = 720 \text{ cm}^3$

Answer

The volume of the triangular prism is $720 \text{ cm}^3$ , which is $720$ cubic centimeters.

Worked example

A prism with a trapezoid base

The base of a right prism is a trapezoid, a four-sided shape with at least one pair of parallel sides. Its parallel sides are 5 ft and 9 ft long, and the perpendicular distance between them is 4 ft. The height of the prism, perpendicular to the base, is 7.5 ft. Find the volume of the prism.

Check the measures. All the lengths are positive and in feet. The 4 ft distance is perpendicular to the parallel sides, and the prism height is perpendicular to the base. The volume will be in cubic feet.
Find the base area. Two copies of a trapezoid fit together to make a parallelogram, so a trapezoid's area is half of the sum of its parallel sides times the perpendicular distance between them.
$B = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 7 \times 4 = 28 \text{ ft}^2$
Multiply the base area by the prism height. A layer 1 ft thick holds 28 cubic feet. The height of 7.5 ft makes 7 full layers and half of another layer.
$V = B \times h = 28 \times 7.5 = 196 + 14 = 210 \text{ ft}^3$

Answer

The volume of the prism is $210 \text{ ft}^3$ , which is $210$ cubic feet.

Takeaway. The volume of a right prism is the number of unit cubes that fill it, and one rule finds it: base area times prism height, $V = B \times h$ . The base area tells you how much a layer 1 unit thick holds, and the height tells you how many layers there are. Only the base area step changes with the shape of the base: a rectangle or square, a triangle, or another shape such as a trapezoid. Use positive lengths in the same unit, use heights that are perpendicular, and give the volume in cubic units.

Next, you will use this rule to find a missing dimension when you know the volume, and you will solve volume word problems.