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\(V\)

Solving Problems with
Volume Formulas

Master the volume formulas for cylinders, pyramids, cones, and spheres and learn to apply them to real-world problems.

Geometry

What Is Volume?

Volume measures the amount of three-dimensional space enclosed by a solid figure. It answers the question: how much can this shape hold?

In this lesson we focus on four families of solids: cylinders, pyramids, cones, and spheres.

Meet the Four Solids

Cylinder
Two parallel circular bases connected by a curved surface
Pyramid
A polygon base with triangular faces meeting at an apex
Cone
A circular base tapering smoothly to a single apex point
Sphere
Every point on the surface is the same distance from the center

Key Dimensions to Know

\(r\)
\(h\)
 Cylinder
\(r\)
\(h\)
 Cone
\(h\)
 Pyramid
\(r\)
 Sphere

The Volume Formulas

Cylinder
$$V = \pi r^2 h$$
Cone
$$V = \tfrac{1}{3}\pi r^2 h$$
Pyramid
$$V = \tfrac{1}{3}Bh$$
Sphere
$$V = \tfrac{4}{3}\pi r^3$$

The \(\frac{1}{3}\) Relationship

\(\pi r^2 h\)
 Cylinder = 3 times
\(\tfrac{1}{3}\pi r^2 h\)
 Cone

Comparing Shapes with Same \(r\) and \(h\)

Solid Formula If \(r=3, h=6\)
Cylinder \(\pi r^2 h\) \(54\pi \approx 169.6\)
Cone \(\frac{1}{3}\pi r^2 h\) \(18\pi \approx 56.5\)
Sphere \(\frac{4}{3}\pi r^3\) \(36\pi \approx 113.1\)

How to Solve Volume Problems

1
Identify the shape. Is it a cylinder, cone, pyramid, or sphere?
2
Write the formula. Select the correct volume formula for that shape.
3
Extract the dimensions. Find \(r\), \(h\), or \(B\) from the diagram or problem text. Convert units if needed.
4
Substitute and compute. Plug values into the formula and simplify. Leave in terms of \(\pi\) or round as directed.
5
State the answer with units. Always include cubic units: \(\text{cm}^3\), \(\text{in}^3\), etc.

Example 1: Cylinder Volume

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

\(5 \text{ ft}\)
\(12 \text{ ft}\)

Example 2: Cone Volume

A cone-shaped funnel has a radius of 4 cm and a height of 9 cm. What is its volume?

\(4 \text{ cm}\)
\(9 \text{ cm}\)

Example 3: Pyramid Volume

A square pyramid has a base side length of 6 m and a height of 10 m. Find the volume.

\(10 \text{ m}\)
Base: \(6 \text{ m} \times 6 \text{ m}\)

Example 4: Sphere Volume

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

\(12 \text{ cm}\)
diameter = 24 cm, so \(r = 12 \text{ cm}\)

Example 5: Composite Solid

A silo is shaped like a cylinder topped with a hemisphere. The radius is 7 ft and the cylinder height is 20 ft. Find the total volume.

\(7 \text{ ft}\)
\(20 \text{ ft}\)

Common Mistakes to Avoid

✕ Using Diameter for Radius
Formulas use \(r\), not \(d\). If given a diameter, always divide by 2 before substituting.
✕ Forgetting the \(\frac{1}{3}\)
Cones and pyramids need the \(\frac{1}{3}\) factor. Skipping it gives 3 times the correct answer.
✕ Confusing Height and Slant
Volume uses the perpendicular height, not the slant height along the surface.

Formula Summary

Solid Volume Formula Key Variable
Cylinder \(V = \pi r^2 h\) \(r\), \(h\)
Cone \(V = \frac{1}{3}\pi r^2 h\) \(r\), \(h\)
Pyramid \(V = \frac{1}{3}Bh\) \(B\) (base area), \(h\)
Sphere \(V = \frac{4}{3}\pi r^3\) \(r\) only
Hemisphere \(V = \frac{2}{3}\pi r^3\) \(r\) only

Question 1 of 5

A cylinder has radius \(r = 4 \text{ cm}\) and height \(h = 10 \text{ cm}\). What is its volume?

Question 2 of 5

A cone has a diameter of \(10 \text{ in}\) and a height of \(12 \text{ in}\). What is its volume?

Question 3 of 5

A sphere has a volume of \(288\pi \text{ cm}^3\). What is its radius?

Question 4 of 5

A rectangular pyramid has a base of \(8 \text{ m}\) by \(5 \text{ m}\) and a height of \(15 \text{ m}\). What is its volume?

Question 5 of 5

A grain silo is a cylinder (\(r = 6 \text{ ft}\), \(h = 30 \text{ ft}\)) topped with a cone (same radius, cone height = \(8 \text{ ft}\)). What is the total volume?