Volume of Prisms and Pyramids | Grade 8 Math | LevelKite

You have already unfolded solids into nets, which show every face laid out flat. A net is about the outside of a solid. Now you will measure the space inside.

Volume is the amount of space a solid takes up. You measure it in cubic units. One cubic centimeter, written $1\text{ cm}^3$ , is the space that a cube with $1\text{ cm}$ edges takes up. A volume of $60\text{ cm}^3$ means the solid takes up as much space as $60$ of those cubes.

In this lesson you will find the volume of two kinds of solids, prisms and pyramids. You will learn what each one is, then see why two measurements are enough to find its volume.

A polygon is a flat, closed shape with straight sides, such as a triangle or a rectangle.

A prism is a solid with two identical polygon faces, called bases, that are parallel (always the same distance apart). The other faces are flat and join the matching sides of the two bases. Slice a prism parallel to its bases, and every slice has the same shape and size as a base.

Lines and flat surfaces are perpendicular when they meet at a right angle, $90^\circ$ . The height of a prism is the perpendicular distance between its two bases. If a side edge is tilted, it is longer than the height, so do not use it as the height.

Here is why base area and height give a prism's volume. Think of a box: a prism whose base is a rectangle $4\text{ cm}$ long and $3\text{ cm}$ wide, with side edges perpendicular to the base. A rectangle's area is its length times its width, so the base area is $4 \times 3 = 12\text{ cm}^2$ . That means $12$ cubes with $1\text{ cm}$ edges cover the base in a single layer $1\text{ cm}$ tall. Each extra centimeter of height adds one more identical layer.

A box with a rectangular base 4 cm long and 3 cm wide

Height in cm	Cubes that fill the box	Volume in cubic cm
$1$	$1$ layer of $12$ cubes	$12 \times 1 = 12$
$2$	$2$ layers of $12$ cubes	$12 \times 2 = 24$
$3$	$3$ layers of $12$ cubes	$12 \times 3 = 36$
$5$	$5$ layers of $12$ cubes	$12 \times 5 = 60$

In every row, the volume is the base area times the height. The base area tells you how many cubic units fill a layer one unit tall. The height tells you how many of those layers there are.

Let $B$ stand for the area of one base, $h$ for the perpendicular height and $V$ for the volume. All three are positive numbers with units. Then for a prism, $V = B \times h$ .

This is true for every prism, not only for boxes with whole-number measurements. Every slice parallel to the base matches the base, so each unit of height still adds $B$ cubic units.

Units matter. Put every length in the same unit before you multiply. Then $B$ is in square units, $h$ is in that length unit and $V$ is in cubic units. For example, $\text{cm}^2 \times \text{cm} = \text{cm}^3$ .

Worked example

Volume of a triangular prism

A prism has two identical triangular bases. In each triangle, one side is $10\text{ cm}$ long, and the triangle's height perpendicular to that side is $6\text{ cm}$ . The two bases are $15\text{ cm}$ apart, measured perpendicular to the bases. Find the volume of the prism.

Sort the measurements. The 10 cm side and the 6 cm triangle height both lie in the base, so they give the base area. The 15 cm distance between the bases is perpendicular to them, so it is the prism's height. Every length is in centimeters, so the units already match.
$h = 15\text{ cm}$
Find the base area. A triangle's area is half of a side length times the height perpendicular to that side.
$B = \frac{1}{2} \times 10 \times 6 = \frac{1}{2} \times 60 = 30\text{ cm}^2$
Multiply the base area by the prism's height. Every slice parallel to the base is the same triangle, so each centimeter of height adds 30 cubic centimeters.
$V = B \times h = 30 \times 15 = 450\text{ cm}^3$

Answer

The volume of the triangular prism is $450\text{ cm}^3$ .

A pyramid is a solid with one polygon base. All of its other faces are triangles that meet at a single point called the apex.

The height of a pyramid is the perpendicular distance from the apex to the base. Picture the pyramid resting on its base on a table: the height is how far the apex is above the tabletop, measured at a right angle to the tabletop.

A pyramid also has slanted lengths. A slant height runs along a triangular face, from the apex to a base edge, and meets that edge at a right angle. When the face is tilted, its slant height is longer than the pyramid's height. A problem may state both lengths, so read which one is perpendicular to the base.

Slices of a pyramid parallel to its base get smaller toward the apex. So a pyramid takes up less space than a prism with the same base and height. A cube shows exactly how much less.

Think of a cube with $6\text{ cm}$ edges. A cube is a prism with a square base, so its volume is base area times height: $(6 \times 6) \times 6 = 216\text{ cm}^3$ . Now join the center of the cube to all eight corners. This cuts the cube into $6$ identical pyramids. Each face of the cube is the base of one pyramid, and the center is the apex of all six. So each pyramid has volume $216 \div 6 = 36\text{ cm}^3$ .

Look at one of those pyramids. Its base is a face of the cube, a square with area $B = 6 \times 6 = 36\text{ cm}^2$ . Its apex is the center of the cube, so its perpendicular height is half an edge: $h = 3\text{ cm}$ . A prism with that base and height would hold $B \times h = 36 \times 3 = 108\text{ cm}^3$ . One third of that matches the pyramid's volume exactly:

\frac{1}{3} \times B \times h = \frac{1}{3} \times 108 = 36\text{ cm}^3

Key concept

Volume from base area and perpendicular height

The volume of a prism or a pyramid comes from two measurements: the area of its base and its perpendicular height.

Let $B$ be the area of the base in square units, $h$ the perpendicular height in the matching length unit and $V$ the volume in cubic units. Every length is positive and in the same unit.

Prism: $h$ is the perpendicular distance between the two bases, and $V = B \times h$ .
Pyramid: $h$ is the perpendicular distance from the apex to the base, and $V = \frac{1}{3} \times B \times h$ . The cube showed this for one pyramid, and the same one-third fraction is true for every pyramid.

Find $B$ first from the shape and dimensions of the base. Never use a slant height in place of $h$ .

V_{\text{prism}} = B \times h \qquad V_{\text{pyramid}} = \frac{1}{3} \times B \times h

Worked example

Volume of a square pyramid

A pyramid has a square base with edges $6\text{ m}$ long. Its apex is $4\text{ m}$ above the base, measured perpendicular to the base. The slant height along each triangular face is $5\text{ m}$ . Find the volume of the pyramid.

Choose the height. The rule needs the perpendicular distance from the apex to the base, which is 4 m. The 5 m slant height runs along a tilted face, so you do not use it. Every length is in meters, so the units already match.
$h = 4\text{ m}$
Find the base area. The base is a square, so its area is the edge length times itself.
$B = 6 \times 6 = 36\text{ m}^2$
Take one third of the base area times the height, because a pyramid holds one third as much as a prism with the same base and height. Multiplying by one third is the same as dividing by 3, so multiply 36 by 4, then divide by 3.
$V = \frac{1}{3} \times 36 \times 4 = \frac{144}{3} = 48\text{ m}^3$

Answer

The volume of the square pyramid is $48\text{ m}^3$ .

Worked example

Mixed units and an exact answer

A pyramid has a rectangular base $10\text{ cm}$ long and $5\text{ cm}$ wide. Its apex is $0.2\text{ m}$ above the base, measured perpendicular to the base. Find the volume in cubic centimeters. Give the exact value, then round to the nearest tenth.

Match the units first. The base is in centimeters but the height is in meters, and multiplying them as they are would not give cubic centimeters. Because 1 m equals 100 cm, multiply the number of meters by 100. The length stays the same; only its unit changes.
$h = 0.2 \times 100 = 20\text{ cm}$
Find the area of the rectangular base: length times width.
$B = 10 \times 5 = 50\text{ cm}^2$
The solid is a pyramid, so take one third of the base area times the perpendicular height.
$V = \frac{1}{3} \times 50 \times 20 = \frac{1}{3} \times 1000 = \frac{1000}{3}\text{ cm}^3$
This fraction is the exact volume. Dividing 1000 by 3 gives 333.333..., a decimal that never ends, so the rounded value is an approximation.
$V = \frac{1000}{3}\text{ cm}^3 \approx 333.3\text{ cm}^3$

Answer

The volume of the pyramid is exactly $\frac{1000}{3}\text{ cm}^3$ , which is about $333.3\text{ cm}^3$ when rounded to the nearest tenth.

Takeaway. Volume measures how many cubic units fill a solid. For a prism, multiply the area of the base by the perpendicular height, because every unit of height adds as many cubic units as the base has square units. For a pyramid, take one third of the base area times the perpendicular height.

Each time, work out the base area from the shape of the base, put every length in the same unit, use the perpendicular height and never a slant height, and write the answer in cubic units.

Next, you will find the surface area of prisms and pyramids, which measures the outside of these solids instead of the space inside.