Cone Volume and Surface Area Calculator

How to use the cone volume and surface area calculator

Start with the values you know

For a forward calculation, enter the circular base radius and the perpendicular height. To work backwards, combine volume with height to recover radius, or combine volume with radius to recover height. Both inverse paths are closed forms, so the result does not depend on repeated numerical guessing.

1. 1

   Radius and perpendicular height

   Use this pair to calculate volume directly, then derive the base area, slant height, lateral area, total surface area, and capacity conversions.

2. 2

   Volume and perpendicular height

   Use this pair when capacity and vertical height are fixed. The calculator recovers radius with r = √(3V ÷ (πh)).

3. 3

   Volume and radius

   Use this pair when the opening or base is fixed. The calculator recovers perpendicular height with h = 3V ÷ (πr²).

A cone occupies exactly one third of the volume of a cylinder with the same circular base and perpendicular height. The factor is not an approximation: the cylinder volume is πr²h, while the cone volume is πr²h ÷ 3.

Keep the two heights separate

The perpendicular height runs at right angles to the base and belongs in the volume formula. The slant height runs from the apex along the surface to the rim and belongs in the lateral-area formula. For a non-degenerate right cone, the slant height is longer than the perpendicular height.

A frustum has its apex removed and therefore has two circular radii and a shorter perpendicular height. Its volume is not obtained by treating it as a complete cone with either end radius. This page calculates a complete right circular cone, not a frustum.

Why the one-third factor matters

Keeping the radius fixed makes the cone and cylinder share the same base area. Keeping the perpendicular height fixed gives them the same base-to-top distance. Under those two conditions, the cone’s volume is exactly one third of the cylinder’s volume.

Capacity and unit reporting

The working length unit may be mm, cm, m, in, ft, or yd. Volume is reported in the corresponding cubic unit and converted to litres, US gallons, imperial gallons, and cubic feet without changing the underlying cone dimensions.

The engine stores 3.785411784 litres per US gallon, 4.54609 litres per imperial gallon, and 28.316846592 litres per cubic foot directly. These literals preserve the exact published definitions instead of rebuilding them through floating-point arithmetic that can introduce tiny representation artefacts.

Surface area of a cone, and the sector it unrolls into

Cut a cone from rim to apex and flatten it out. The curved surface is not a triangle — it is a sector of a circle. The sector's radius is the slant height, because the slant is the distance from apex to rim and flattening does not change it. The sector's arc is the base circumference 2πr, because that edge used to be the rim.

Those two facts give the formula with no calculus in sight. A sector of radius l whose arc measures 2πr has area ½ × l × 2πr, which is πrl. Add the base, a plain circle of area πr², and the total surface area is πr² + πrl.

This is the real reason the slant belongs in the area formula while the vertical height does not. A sector is fixed by exactly two things — its radius, which here is the slant, and its arc, which here is the base circumference. The perpendicular height is neither of them.

So substituting it for the slant supplies the wrong quantity rather than a slightly wrong number. The height is not lost, mind: the arc gives back the radius, and from the radius and slant it returns as √(l² − r²). It is simply not what the area formula asks for.

Where the third actually comes from

The one-third is not a convention or a measured approximation. It falls out of summing the areas of the circular slices from apex to base, each of which is smaller than the last in proportion to the square of its distance from the tip.

That squaring is the whole reason the factor is a third rather than a half. A shape whose cross-section shrank linearly would hold half its enclosing prism; because a circle's area shrinks with the square of its radius, the cone keeps only a third.

The same squaring decides how carefully to measure. A radius one per cent high lifts the volume by 2.01 per cent, while a height one per cent high lifts it by only one per cent. On the worked cone of radius 3 and height 4, holding 12π, a radius read as 3.03 returns 38.46 cubic units against a true 37.70.

The formula, worked line by line

Let r be the circular base radius, h the perpendicular height, l the slant height, V the volume, Aᵦ the base area, and Aₗ the lateral area.

The radius and perpendicular height form the right-angled cross-section used to calculate slant height. All input lengths must use the same working unit.

V = πr²h ÷ 3
r = √(3V ÷ (πh))
h = 3V ÷ (πr²)
Aᵦ = πr²
l = √(r² + h²)
Aₗ = πrl
Aₜ = πr² + πrl
Vexact = qπ when r²h ÷ 3 reduces to a tidy rational coefficient q

The inverse radius formula contains both a factor of 3 and a square root. The factor reverses the cone’s one-third volume rule, while the square root reverses the squared radius.

Slant height is derived after radius and perpendicular height are known. It is used for the sloping surface area but must not replace perpendicular height in the volume formula.

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