Cone Volume Calculator: Learn the (1/3)πr²h Rule
ByMuhammad Ali•Founder of KruskalCode
13:01
6 min read
Cone volume questions appear constantly in UK and US school geometry: traffic cones, paper cups, and lab glassware are all modelled as right circular cones. Once you measure the base radius and the perpendicular height in the same unit, one compact formula gives the interior space in cubic units.
Explanation
A right circular cone has a flat circular base of radius r and a single apex directly above the centre of that circle. The perpendicular height h is the vertical drop from apex to base, not the slant distance down the side. The volume counts three-dimensional space inside, so area at the base (πr²) is scaled by height and then reduced by the tapering shape, giving the classical factor of one-third.
Formula
Let r be the base radius and h the perpendicular height. Then V = (1/3)πr²h. If only the diameter d is known, substitute r = d/2 before squaring.
Example
Suppose r = 5 inches and h = 12 inches. Then r² = 25 and V = (1/3)π × 25 × 12 = 100π ≈ 314.16 in³. In centimetres the same numeric idea applies: r = 5 cm and h = 12 cm would give about 314.16 cm³—always keep length units consistent.
How to use the related calculator
Open the Cone Volume Calculator page, type a positive radius and a positive perpendicular height using the same length unit throughout, then read the output lines. They restate the formula, echo your inputs, give V in cubic units, and remind you that a cylinder sharing the same r and h would be three times larger—useful when you revise related solid problems.
Try the related calculator
Open toolFAQ
What is the difference between height and slant height?
Height h is measured straight down from the tip to the base centre. Slant height runs along the curved surface from the tip to the rim. This volume tool uses h; if a problem only gives slant height, use Pythagoras with r to find h first.
Does the formula work for oblique cones?
Yes for the standard right circular model taught in school: the same V = (1/3)πr²h applies when the apex lies above the base centre. Highly tilted real-world shapes need more advanced modelling beyond this introductory calculator.
Can I compare this with cylinder and sphere tools on the site?
Absolutely. Check a cylinder with matching r and h to see the factor of three, and compare with sphere volume when you revise mixed solid-geometry worksheets.
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About the author
Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.
Need a custom calculator, dashboard, or automation workflow? Reach out to KruskalCode.
