Cone Volume Calculator - Calculate Cone & Frustum Volume

What is a cone volume calculator?

A cone volume calculator finds the volume of a cone, or of a truncated cone (a frustum), from its radius and height. A cone is a solid shape with one flat, circular base that narrows smoothly to a single point, called the apex. A frustum is what remains of a cone after the top has been sliced off by a cut parallel to the base, leaving two circular faces instead of one.

Cone volume formula

The volume of a full cone depends on the radius of its base and its height.

$V = \frac{1}{3} \times \pi \times r² \times h$

• V is the volume of the cone.
• r is the radius of the circular base.
• h is the height, measured straight from the base to the apex.
• π (pi) is approximately 3.14159.

Worked example

A cone with a base radius of 3 units and a height of 4 units has this volume:

V = (1/3) × π × 3² × 4 = (1/3) × π × 36 = 12π ≈ 37.70 cubic units

How to calculate cone volume with the calculator

The calculator takes three inputs:

1. Radius — the radius of the base.
2. Height — the height of the cone, from base to apex.
3. Cut height from base (optional) — how far above the base the cone is cut, if the top is sliced off.

Leaving the cut height blank, or entering 0, gives the volume of the full, uncut cone. Entering a cut height smaller than the full height gives the volume of the frustum below the cut. The diagram below the inputs marks the same three measurements on the shape.

The calculator rejects a radius or height that is zero, negative, blank, or too large to be handled as an ordinary number. In those cases it shows a notice instead of a volume, so it never displays a made-up figure.

Truncated cone (frustum) volume formula

A frustum's volume depends on the radii of its two circular faces and its own height.

$V = \frac{1}{3} \times \pi \times h \times (R² + r² + R \times r)$

• R is the radius of the larger, bottom face.
• r is the radius of the smaller, top face.
• h is the height of the frustum, measured between the two faces.

The calculator does not ask for the top radius directly. Instead, it takes the base radius, the height the cone would have if it were not cut, and the cut height. It then works out the top radius using similar triangles:

$r = R \times \left(1 - \frac{c}{f}\right)$

where c is the cut height above the base and f the full height of the uncut cone.

Worked example

Suppose the base radius is 3 units, the full (uncut) cone height is 12 units, and the cut height is 4 units.

Top radius = 3 × (1 − 4/12) = 3 × 2/3 = 2 units

V = (1/3) × π × 4 × (3² + 2² + 3 × 2) = (1/3) × π × 4 × 19 = (76/3)π ≈ 79.59 cubic units

If the cut height is equal to or greater than the full height, the cut falls at or above the apex, so nothing is removed. The calculator then returns the volume of the full cone, and the diagram draws the full cone.

History of the cone volume formula

Ancient Greek mathematicians were the first to link a cone's volume to that of a cylinder. Democritus, active in the late 5th century BCE, is credited with the claim that a cone's volume equals one third of a cylinder sharing the same base and height. He did not prove it. Eudoxus of Cnidus, working in the following century, gave the first rigorous proof using a technique called the method of exhaustion, an early ancestor of integral calculus. The formula V = (1/3)πr²h has stayed the same ever since.

Uses of cone volume calculations

Cone and frustum volumes come up whenever a shape narrows from a wide base to a point or a smaller top. Engineers use the formulas to size conical tanks, hoppers, and funnels. Architects use them to estimate materials for conical roofs. Geologists use them to estimate the volume of volcanic cones and similar landforms.

Frequently asked questions

How do you calculate the volume of a cone?

Multiply π by the square of the radius, multiply the result by the height, then divide by 3: V = (1/3)πr²h. A cone with a 5 cm radius and a 10 cm height has a volume of about 261.80 cubic centimeters.

What is the formula for a truncated cone (frustum)?

V = (1/3)πh(R² + r² + Rr), where R and r are the radii of the two circular faces and h is the frustum's own height, measured between those faces.

How does the calculator find the top radius of a frustum?

It uses similar triangles. The top radius equals the base radius multiplied by (1 − cut height ÷ full cone height). This assumes the frustum comes from a straight-sided cone.

What happens if the cut height is greater than or equal to the height?

The calculator treats the shape as a full, uncut cone and returns V = (1/3)πr²h using the entered radius and height.

What happens if radius or height is zero or negative?

The calculator requires both the radius and the height to be greater than zero. If either is zero, negative, blank, or too large to be handled as an ordinary number, it shows a notice instead of a volume.

How do you find cone volume from the diameter instead of the radius?

Divide the diameter by 2 to get the radius, then apply V = (1/3)πr²h as usual.

References

1. Weisstein, Eric W. "Cone." MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/Cone.html
2. Weisstein, Eric W. "Frustum." MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/Frustum.html