Cone Calculator
Volume by vertical height, surface area by slant height.
Work out Cone. Volume by vertical height, surface area by slant height. Exact formulas, checked against worked examples.
Vertical height, not the sloping edge.
Surface area needs the slant height, not the vertical height. If your measurement was taken along the sloping side rather than straight up the axis, enter it differently — the two are only equal for a cone of zero radius.
Volume (units³)
37.6991
Slant height 5 units · surface 75.3982 units²
The distinction that trips people: volume uses the vertical height (4), surface area uses the slant height (5). Using the wrong one gives a plausible answer that is simply wrong. The slant is √(r² + h²) — the hypotenuse of the triangle you get by slicing the cone in half. A cone holds exactly a third of the cylinder around it: 37.6991 against 113.0973.
How the Cone Calculator works
The distinction that catches people: volume uses the vertical height, surface area uses the slant height. Both appear in the same problem, and using the wrong one gives a plausible answer that is simply wrong. A cone holds exactly a third of the cylinder around it.
Also known as: volume of a cone · cone surface area calculator · slant height calculator · cone volume formula
Two heights, and why it matters
A cone has a vertical height, straight up the axis, and a slant height, down the sloping side to the rim. They are related by √(r² + h²) — the hypotenuse of the triangle you get by slicing the cone in half.
Volume uses the vertical height. Surface area uses the slant. Both appear in the same problem, and picking the wrong one produces a plausible answer rather than an obvious error, which is exactly what makes it the commonest mistake with this shape.
In practice
A radius of 3 and a height of 4 give a slant height of exactly 5 — a 3-4-5 triangle standing on its side.
The volume is 12π, or 37.70 cubic units. The curved surface is πrl = 15π and the base is 9π, totalling 24π or 75.40 square units.
The cylinder around that cone holds 36π. The cone holds exactly a third of it, which is not an approximation: three cones of matching radius and height fill their cylinder precisely.
Frequently asked questions
How do I find the volume of a cone?
One third of the base area times the vertical height. A radius of 3 and a height of 4 give 12π, or 37.70 cubic units.
What is slant height?
The distance from the apex down the sloping side to the rim — √(r² + h²). For a radius of 3 and height of 4 it is exactly 5, and it is what surface area depends on.
Why is a cone one third of a cylinder?
It follows from integrating the shrinking circular cross-sections, and it is the same one third that governs a pyramid inside its prism. Three cones of matching radius and height fill their cylinder exactly.
Which height did I measure?
If you measured straight up the axis, that is the vertical height. If you measured along the sloping side, that is the slant. They differ, and the page uses each where it belongs.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/cone-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="Cone Calculator"></iframe>The only condition is that the credit line stays visible. It sits inside the frame, so you do not have to do anything to keep it.