Volume of a Cone

Statement

The volume of a cone is given by:

\[ V = \tfrac{1}{3}\pi r^2 h \]

where \(r\) is the radius of the circular base and \(h\) is the perpendicular height of the cone.

Why it’s true

• A cone is like a pyramid with a circular base.
• For any pyramid, volume = \(\tfrac{1}{3}\)(base area)(height).
• The base area of a cone is \(\pi r^2\), so the formula becomes \(\tfrac{1}{3}\pi r^2 h\).

Recipe (how to use it)

1. Identify the radius \(r\) of the cone’s base.
2. Find the perpendicular height \(h\) (not the slant height).
3. Square the radius: \(r^2\).
4. Multiply by \(\pi\), then by the height \(h\).
5. Divide the result by 3.

Spotting it

Look for problems mentioning cones or pyramids with circular bases, often in 3D geometry questions.

Common pairings

• Surface area of a cone (different formula).
• Pythagoras used when only slant height is given, to find the perpendicular height.

Mini examples

1. Given: \(r=3\), \(h=9\). Find: Volume. Answer: \(\tfrac{1}{3}\pi \cdot 9 \cdot 9 = 27\pi\).
2. Given: \(r=5\), \(h=12\). Find: Volume. Answer: \(\tfrac{1}{3}\pi \cdot 25 \cdot 12 = 100\pi\).

Pitfalls

• Using the slant height instead of the perpendicular height.
• Forgetting to divide by 3 (cone is one third of a cylinder).
• Not squaring the radius.

Exam strategy

• Write down the formula before substituting values.
• If height is missing but slant height is given, use Pythagoras with radius.
• Leave answers in terms of \(\pi\) unless a decimal is requested.

Summary

The volume of a cone is one third of the volume of a cylinder with the same base and height. Always use the perpendicular height and take care with squaring the radius.