Volume of a Cone
In short: Volume of a cone is a third of the volume of the cylinder that has the same base and height: V = (1/3)πr²h.
8th GradeGeometry8.G.C.9
What is Volume of a Cone?
Scoop sand with a cone-shaped paper cup into a matching cylinder, same width and same height, and three scoops fill it exactly, so the volume of a cone is one third of that cylinder’s volume.
With base radius \(r\) and vertical height \(h\), \(V = \frac{1}{3}\pi r^2 h\). Use the height measured straight down from the tip to the center of the base, not the slanted side.
How to spot it
- A solid has a circular base and tapers to a single point, like a funnel, traffic cone or party hat.
- Two solids with equal radius and equal height are compared, one pointed and one flat-topped.
- You are given a slant length as well as a height, and must pick the right one for the formula.
- The word capacity or how much it holds appears next to a cone-shaped container.
The method
- Identify the radius of the circular base, halving the diameter if needed.
- Find the perpendicular height from the tip to the base, using the Pythagorean theorem if only the slant is given.
- Compute \(\pi r^2 h\), then divide by 3 (or multiply by \(\frac{1}{3}\)).
- Write the answer in cubic units, leaving π in the answer or rounding at the end.
Volume of a Cone: solved example
Problem
A school lab funnel is shaped like a cone with rim radius 5 inches and depth 12 inches. Round its capacity to the nearest tenth of a cubic inch.
Solution
Substitute into the formula: \(V = \frac{1}{3}\pi (5)^2 (12) = \frac{1}{3}\pi \cdot 300 = 100\pi\).
Using \(\pi \approx 3.14159\), \(100\pi \approx 314.2\). The funnel holds about 314.2 cubic inches.
Try it
Question 1
Find the volume of a cone with radius 6 cm and height 7 cm, in terms of π.
Show the answer
\(\frac{1}{3}\pi \cdot 36 \cdot 7 = 84\pi\) cm³, about 263.9.
Question 2
A drum holds 90 cubic feet. A pointed hopper over the same circle with equal height is built. How much does the hopper hold?
Show the answer
One third of 90, so 30 cubic feet.
Volume of a Cone vs. Volume of a Cylinder
Both formulas contain \(\pi r^2 h\). The cylinder keeps all of it; this solid keeps only a third because it narrows to a point. With equal base and height, the cylinder always holds three times as much.
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Whole dictionary: A–Z · 8th Grade · Geometry