Apply Volume Formulas

Master the formulas for volumes of cones, cylinders, and spheres and apply them to solve real-world problems.

8th Grade Math

What Is Volume?

Volume measures the amount of three-dimensional space an object occupies. We express volume in cubic units such as \(\text{cm}^3\), \(\text{in}^3\), or \(\text{ft}^3\).

In this lesson you will learn the formulas for three important 3-D shapes: cylinders, cones, and spheres.

Meet the Three Shapes

Cylinder

Two parallel circular bases connected by a curved surface. Like a soup can.

Cone

One circular base that tapers to a point (apex). Like a party hat.

Sphere

Every point on its surface is the same distance from the center. Like a basketball.

Radius and Height

Volume of a Cylinder

Volume of a Cone

Volume of a Sphere

Cone vs. Cylinder: the \(\frac{1}{3}\) Factor

How to Apply a Volume Formula

1

Identify the shape: Is it a cylinder, cone, or sphere?

2

Write the formula: \(V = \pi r^2 h\), \(V = \frac{1}{3}\pi r^2 h\), or \(V = \frac{4}{3}\pi r^3\).

3

Check for radius vs. diameter: If given a diameter, divide by 2 to get \(r\).

4

Substitute and compute: Plug in values, simplify step by step.

5

Label your answer with correct cubic units (e.g., \(\text{cm}^3\)).

Example 1: Cylinder

A cylindrical water tank has a radius of 5 ft and a height of 12 ft. Find its volume.

Example 2: Cone

An ice-cream cone has a diameter of 6 cm and a height of 10 cm. Find its volume.

Example 3: Sphere

A basketball has a diameter of 24 cm. What is its volume?

Example 4: Combined Shape

A silo is a cylinder topped with a cone. The radius is 4 m, the cylinder height is 10 m, and the cone height is 3 m. Find the total volume.

Common Mistakes to Avoid

✕ Using Diameter as Radius

If a problem gives diameter = 10, the radius is 5. Always divide by 2 first.

✕ Forgetting \(\frac{1}{3}\)

Cone volume is one-third of a cylinder. Omitting \(\frac{1}{3}\) gives an answer 3 times too large.

✕ Squaring vs. Cubing

Cylinders and cones use \(r^2\). Spheres use \(r^3\). Mixing these up changes the answer drastically.

Summary Table

	Cylinder	Cone	Sphere
Formula	\(\pi r^2 h\)	\(\frac{1}{3}\pi r^2 h\)	\(\frac{4}{3}\pi r^3\)
Uses height?	Yes	Yes	No
Power of \(r\)	\(r^2\)	\(r^2\)	\(r^3\)
Fraction?	None	\(\frac{1}{3}\)	\(\frac{4}{3}\)
Relationship	Base shape	\(\frac{1}{3}\) of cylinder	Unique

Question 1 of 5

A cylinder has a radius of 3 cm and a height of 7 cm. What is its volume?

\(V = \pi r^2 h = \pi(3)^2(7) = \pi(9)(7) = 63\pi\). Remember: square the radius first, then multiply by the height.

Question 2 of 5

A cone has a diameter of 10 in and a height of 9 in. What is its volume?

Diameter = 10, so \(r = 5\). Then \(V = \frac{1}{3}\pi(5)^2(9) = \frac{1}{3}\pi(225) = 75\pi\). Don't forget to halve the diameter and include the \(\frac{1}{3}\) factor.

Question 3 of 5

A sphere has a radius of 6 m. What is its volume?

\(V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi\). Cube the radius: \(6^3 = 216\), then multiply by \(\frac{4}{3}\).

Question 4 of 5

A cone and a cylinder have the same radius and height. The cylinder's volume is \(150\pi\). What is the cone's volume?

A cone is one-third the volume of a cylinder with the same base and height. \(\frac{1}{3} \times 150\pi = 50\pi\).

Question 5 of 5

A water tower is a sphere with a diameter of 20 ft. What is its approximate volume? (Use \(\pi \approx 3.14\))

Diameter = 20, so \(r = 10\). \(V = \frac{4}{3}\pi(10)^3 = \frac{4}{3}(3.14)(1000) = \frac{4}{3}(3140) \approx 4{,}186.7\;\text{ft}^3\). Watch out: choice (a) uses the diameter as the radius.