What is the formula for the volume of a cone?

The volume of a cone is V = (1/3)πr²h, where r is the base radius and h is the vertical height. Enter both and the calculator returns the volume instantly. For example, with r = 5 and h = 12 the volume is 100π ≈ 314.16 cubic units.

Why do you divide by 3 in the cone volume formula?

A cone occupies exactly one-third of the cylinder that has the same base and height. The full cylinder holds πr²h, and because the cone narrows from that base to a single point, it contains a third as much — hence the 1/3 in (1/3)πr²h. The ratio comes from integrating the cone's circular cross-sections and is exact, not rounded.

How do I find the slant height of a cone?

The slant height l is the distance along the sloped side from the base rim to the apex, and it is longer than the vertical height h. By the Pythagorean theorem, l = √(r²+h²). For r = 5 and h = 12, l = √(25+144) = √169 = 13. You need l — not h — to compute the lateral surface area πrl.

What units does the cone volume calculator use?

It is unit-agnostic. Enter the radius and height in the same unit and the volume comes out cubed: centimeters give cubic centimeters (cm³), meters give cubic meters (m³), inches give cubic inches. The tool does not convert units, so keep r and h in one consistent unit.

What is the difference between the volume of a cone and a cylinder?

A cone and a cylinder that share the same base radius and height are directly related: the cone's volume is exactly one-third of the cylinder's. The cylinder holds πr²h, while the cone holds (1/3)πr²h, so three identical cones would fill one matching cylinder. This tool computes the cone; multiply the result by 3 to get the cylinder.

Cone Volume Calculator | Free Online Tool

How to use

1.Type the base radius (r) into the first box, in any unit you like.
2.Type the height (h) into the second box — the vertical height from the base to the tip, not the slant height.
3.Read the cone volume instantly below; it also shows the base area, slant height, and lateral surface area.

About Cone Volume Calculator

The volume of a cone is V = (1/3)πr²h, where r is the base radius and h is the height measured straight up from the center of the base to the tip. This cone volume calculator returns that value the instant you enter r and h, and it also shows the base area (πr²), the slant height (√(r²+h²)), and the lateral surface area (πrl) so you can check every part of the shape at once.

Why the one-third? A cone sits inside the cylinder that shares its base and height, and it fills exactly one-third of that cylinder. The cylinder's volume is base area times height, πr²h; the cone tapers to a single point at the top, so it holds a third as much: (1/3)πr²h. This 3-to-1 ratio is not an approximation — it falls straight out of integrating the shrinking circular cross-sections from base to apex, and it holds for any cone, tall or squat.

The two inputs are the base radius r and the vertical height h. Note that h is the perpendicular height, not the slant height. The slant height l is the distance along the sloping side from the rim to the tip, and by the Pythagorean theorem l = √(r²+h²). You need the slant height, not h, to find the lateral (side) surface area πrl or to cut a flat template for a cone, which is why this tool reports it alongside the volume.

Radius and height are lengths, so they can be zero or positive but never negative. If either is 0 the cone collapses to a point or a flat disk and the volume is exactly 0; the calculator shows that instead of an error. Keep both inputs in the same unit: enter centimeters and the volume comes out in cubic centimeters (cm³), enter meters and you get cubic meters (m³). The tool never converts units, so the numbers stay exactly (1/3)πr²h.

Cone volume comes up constantly in the real world: the sand or salt in a conical pile, the liquid in an ice-cream cone or a paper cup, the concrete in a pointed pier, the capacity of a hopper or funnel, and countless geometry and calculus problems. Worked example: with r = 5 and h = 12, the base area is 25π ≈ 78.54 and the volume is (1/3)(25π)(12) = 100π ≈ 314.16 cubic units. Everything runs locally in your browser — nothing is uploaded — so the answer is instant and private.

Frequently asked questions

What is the formula for the volume of a cone?: The volume of a cone is V = (1/3)πr²h, where r is the base radius and h is the vertical height. Enter both and the calculator returns the volume instantly. For example, with r = 5 and h = 12 the volume is 100π ≈ 314.16 cubic units.
Why do you divide by 3 in the cone volume formula?: A cone occupies exactly one-third of the cylinder that has the same base and height. The full cylinder holds πr²h, and because the cone narrows from that base to a single point, it contains a third as much — hence the 1/3 in (1/3)πr²h. The ratio comes from integrating the cone's circular cross-sections and is exact, not rounded.
How do I find the slant height of a cone?: The slant height l is the distance along the sloped side from the base rim to the apex, and it is longer than the vertical height h. By the Pythagorean theorem, l = √(r²+h²). For r = 5 and h = 12, l = √(25+144) = √169 = 13. You need l — not h — to compute the lateral surface area πrl.
What units does the cone volume calculator use?: It is unit-agnostic. Enter the radius and height in the same unit and the volume comes out cubed: centimeters give cubic centimeters (cm³), meters give cubic meters (m³), inches give cubic inches. The tool does not convert units, so keep r and h in one consistent unit.
What is the difference between the volume of a cone and a cylinder?: A cone and a cylinder that share the same base radius and height are directly related: the cone's volume is exactly one-third of the cylinder's. The cylinder holds πr²h, while the cone holds (1/3)πr²h, so three identical cones would fill one matching cylinder. This tool computes the cone; multiply the result by 3 to get the cylinder.

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