Volume of Prisms and Cylinders: V = Bh

The volume of prisms and cylinders is the area of the base times the height: $V = Bh$ . For a cylinder, the base is a circle, so $V = \pi r^2 h$ . Volume measures the space inside a solid, in cubic units such as cm³ or ft³.

Before this lesson: You should know Surface Area of Prisms and Cylinders, Area of Triangles and Circumference and Area of Circles.

Volume as Stacked Layers

Think of a prism as a stack of identical layers, each one a copy of the base. One layer that is 1 unit thick holds $B$ cubic units, where $B$ is the area of the base. A prism $h$ units tall has $h$ layers. So the volume is $B \cdot h$ .

Left: a rectangular prism 5 by 3 by 4 built from unit cubes, with the bottom layer shaded; each layer holds 15 cubes and 4 layers make 60 cubic units. Right: a cylinder with radius 3 and height 10 shown as stacked circular layers, each with area 9 pi; 10 layers make 90 pi cubic units. — Each layer is a copy of the base. Volume = base area × number of layers.

Volume Formulas

\text{Prism: } V = Bh \qquad \text{Cylinder: } V = \pi r^2 h

For a rectangular prism, $B = \ell w$ , so $V = \ell wh$ . For a cube with edge $s$ , $V = s^3$ .

Cavalieri’s principle: If two solids have the same height and the same cross-section area at every level, they have the same volume. So $V = Bh$ also works for oblique (slanted) prisms and cylinders. Use the straight-up height, not the slanted edge.

Two stacks of 8 identical coins. The left stack is straight, forming a right cylinder. The right stack leans, forming an oblique cylinder. Both have the same base area and the same height, so they have the same volume. — Leaning the stack doesn’t change the volume: same base, same height.

How to Find Volume

Identify the base. It is the face that repeats through the solid, not always the bottom.
Find the area of the base, $B$ . For a cylinder, $B = \pi r^2$ .
Find the height $h$ , the perpendicular distance between the two bases.
Multiply: $V = Bh$ . Give the answer in cubic units.

Worked Examples

Example 1: A rectangular prism

Find the volume of a box 5 cm long, 3 cm wide and 4 cm tall.

Find the base area: $B = 5 \cdot 3 = 15$
Multiply by the height: $V = 15 \cdot 4$
Multiply: $V = 60$

Answer: 60 cm³

Example 2: A triangular prism

The bases of a prism are right triangles with legs 6 and 8. The prism is 12 units long. Find its volume.

Find the base area (see area of triangles): $B = \tfrac{1}{2} \cdot 6 \cdot 8$
Multiply: $B = 24$
Multiply by the length between the bases: $V = 24 \cdot 12$
Multiply: $V = 288$

Answer: 288 cubic units. The prism lies on its side, but the bases are still the triangles.

Example 3: A cylinder

Find the volume of a cylinder with radius 3 and height 10.

Write the formula: $V = \pi r^2 h$
Substitute: $V = \pi(3)^2(10)$
Square: $V = \pi(9)(10)$
Multiply: $V = 90\pi$
Use a calculator: $V \approx 282.74$

Answer: $90\pi \approx 282.74$ cubic units

Example 4: Find the height

A cylinder has radius 4 in and volume $200\pi$ in³. Find its height.

Write the formula: $200\pi = \pi(4)^2 h$
Square: $200\pi = 16\pi h$
Divide both sides by $16\pi$ : $h = \frac{200}{16}$
Divide: $h = 12.5$

Answer: 12.5 in

Example 5: Word problem with a fish tank

A fish tank is 24 in long, 12 in wide and 16 in tall. How many cubic inches of water does it hold when full? One gallon is 231 in³. About how many gallons is that?

Find the base area: $B = 24 \cdot 12 = 288$
Multiply by the height: $V = 288 \cdot 16$
Multiply: $V = 4608$
Convert to gallons by dividing by 231 (see unit conversions): $4608 \div 231 \approx 19.95$

Answer: 4,608 in³, or about 19.95 gallons

Common Mistakes

Mistake	Example of the error	Fix
Using the diameter in $\pi r^2 h$	$\pi(12)^2(5)$ for a cylinder 12 across	Halve the diameter first: $\pi(6)^2(5)$ .
Using the slanted edge as the height	Using 9 as the height of an oblique cylinder that is 7 tall	The height is the perpendicular distance between the bases.
Mixing up volume and surface area	Answering a “how much water” question with $2\pi rh + 2\pi r^2$	Capacity is volume: $\pi r^2 h$ .
Writing square units	$V = 60$ cm²	Volume uses cubic units: 60 cm³.
Picking the wrong face as the base	Multiplying the 6 by 12 face by 8 for a triangular prism	The base is the face that repeats: the triangle.

Practice Problems

Give answers with $\pi$ exactly, then rounded to the nearest hundredth.

Find the volume of a cube with edge 7 cm.
Find the volume of a box 10 in by 4 in by 2.5 in.
A prism has equilateral triangle bases with side 6 and area $9\sqrt{3}$ . It is 10 units long. Find its volume.
Find the volume of a cylinder with radius 5 and height 8.
Find the volume of a cylinder with diameter 12 and height 5.
An oblique cylinder has radius 3 and height 7. Its slanted side is 9 units long. Find its volume.
A cylinder has height 3 and volume $147\pi$ . Find its radius.
A prism has a regular hexagon base with side 4 and apothem $2\sqrt{3}$ . It is 10 units tall. Find its volume.
A pool is 20 ft long, 10 ft wide and 5 ft deep everywhere. How many cubic feet of water does it hold? One cubic foot is about 7.48 gallons. How many gallons is that?
A can has radius 3.5 cm and height 12 cm. Find its volume. (1 cm³ = 1 mL)

Show answers

Write the formula: $V = s^3$
Cube: $V = 7^3 = 343$ cm³
Multiply: $V = 10 \cdot 4 \cdot 2.5$
Multiply: $V = 100$ in³
Multiply the base area by the length: $V = 9\sqrt{3} \cdot 10$
Multiply: $V = 90\sqrt{3}$
Use a calculator: $V \approx 155.88$
Substitute: $V = \pi(5)^2(8)$
Square: $V = \pi(25)(8)$
Multiply: $V = 200\pi$
Use a calculator: $V \approx 628.32$
Find the radius: $r = 12 \div 2 = 6$
Substitute: $V = \pi(6)^2(5)$
Square: $V = \pi(36)(5)$
Multiply: $V = 180\pi$
Use a calculator: $V \approx 565.49$
By Cavalieri’s principle, use the height 7, not the slanted side: $V = \pi(3)^2(7)$
Square: $V = \pi(9)(7)$
Multiply: $V = 63\pi$
Use a calculator: $V \approx 197.92$
Write the formula: $147\pi = \pi r^2(3)$
Divide both sides by $3\pi$ : $49 = r^2$
Take the positive square root: $r = 7$
Find the base area (see area of regular polygons): $B = \tfrac{1}{2}(2\sqrt{3})(24)$
Multiply: $B = 24\sqrt{3}$
Multiply by the height: $V = 24\sqrt{3} \cdot 10 = 240\sqrt{3}$
Use a calculator: $V \approx 415.69$
Multiply: $V = 20 \cdot 10 \cdot 5$
Multiply: $V = 1000$ ft³
Convert: $1000 \cdot 7.48 = 7480$ gallons
Substitute: $V = \pi(3.5)^2(12)$
Square: $V = \pi(12.25)(12)$
Multiply: $V = 147\pi$
Use a calculator: $V \approx 461.81$ cm³, or about 461.81 mL

FAQ

What is the formula for the volume of a prism?

$V = Bh$ , where $B$ is the area of the base and $h$ is the height. For a box, that is length × width × height.

What is the volume formula for a cylinder?

$V = \pi r^2 h$ . It is the area of the circular base times the height.

What is Cavalieri’s principle?

Two solids with the same height and the same cross-section area at every level have the same volume. That is why slanted prisms and cylinders use $V = Bh$ too.

What is the difference between volume and surface area?

Volume measures the space inside, in cubic units. Surface area measures the outside covering, in square units.

Summary

Volume is the space inside a solid, measured in cubic units.
Prism: $V = Bh$ . Rectangular prism: $V = \ell wh$ . Cube: $V = s^3$ .
Cylinder: $V = \pi r^2 h$ .
The base is the face that repeats, and the height is perpendicular to the bases.
Cavalieri’s principle: oblique solids use the same formulas.

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