Surface Area Of A Cone Calculator

What this calculator does

The surface area of a cone is its whole outside: the flat circular base, plus the curved slanted side that wraps up to the tip. This tool gives you both parts and the total, from the cone's radius and its height.

Type the two in and you have the area of the outside, broken down for you.

Using the calculator

1. Type the radius of the circular base.
2. Type the height, the straight vertical distance from the base up to the tip.
3. Press Calculate.

Both values have to be positive. The tool reports three numbers: the base area, the curved side area, and the total. There is no unit setting, so they come as plain numbers in square units of whatever unit you used.

The formula | base + lateral = πr² + πrℓ

The total surface area of a cone is the base plus the curved side:

total = (π × radius²) + (π × radius × slant height)

The first part, π × radius², is just the area of the circular base. The second, π × radius × slant height, is the curved side, also called the lateral surface. The tool shows you each separately and then the total, which is handy when you only need one of them.

The slant height, and Pythagoras

The curved side depends on the slant height, written ℓ, which is the distance from the edge of the base up the slope to the tip. That is not the same as the vertical height h, which goes straight up the middle.

The two are linked, though. The radius r, the vertical height h, and the slant ℓ form a right-angled triangle, with the slant as the longest side. So by the Pythagorean theorem, ℓ = √(radius² + height²). You give this tool the radius and the vertical height, and it works the slant out for you before finding the area. For more on that right-triangle relationship, see the Pythagorean theorem calculator.

Why the curved side is πrℓ

The curved side looks awkward to measure, until you unroll it. Cut it up the slope and lay it flat, and it becomes a sector, a pizza-slice shape, taken from a circle whose radius is the slant height ℓ. The curved edge of that sector is the base's circumference, 2 × π × radius. Work out the area of that flat sector and it comes to π × radius × ℓ, which is where the lateral formula comes from. A curved surface, turned into an ordinary flat area.

Square units, and the π it uses

Surface area is an area, so the answers are in square units, matched to whatever unit you measured in: a cone in centimetres gives areas in cm². The tool does not attach a unit, so it shows plain numbers. For π it uses 3.141592654, ten significant figures, finer than any real measurement.

A worked example | radius 6, height 8

Say the radius is 6 and the vertical height is 8.

1. Slant height first: ℓ = √(6² + 8²) = √(36 + 64) = √100 = 10. (A tidy 6, 8, 10 right triangle.)
2. Base: π × 6² = π × 36 ≈ 113.10.
3. Curved side: π × 6 × 10 ≈ 188.50.
4. Total: 113.10 + 188.50 ≈ 301.59.

So the outside comes to about 301.59 square units, with the curved side making up most of it. Notice how the slant height fell straight out of Pythagoras.

The skin and the space

This is the cone's outer skin. For the space it encloses, which is a third of the matching cylinder, see the volume of a cone calculator. And for the other round surfaces, see the cylinder and the sphere, or the all-in-one surface area calculator.

Questions people ask

What is the surface area of a cone with radius 6 and height 8?

About 301.59 square units: a base of roughly 113.10 and a curved side of roughly 188.50, using a slant height of 10.

What is the slant height?

The distance from the base's edge up the slope to the tip. It is √(radius² + height²) by the Pythagorean theorem, and the tool works it out from the radius and vertical height.

Is the height the slanted side?

No. The height is the straight vertical distance up the middle. The slanted side is the slant height, which the formula needs and the tool computes for you.

Why is the curved side π × radius × slant height?

Because unrolling it gives a sector of a circle of radius equal to the slant height, and that sector's area works out to π × radius × slant height.

What unit is the answer in?

Whatever unit you used, squared. The tool shows plain numbers, so a cone in centimetres gives cm².

References

A note on where this comes from. The curved side of a cone unrolls into a sector of a circle, giving a lateral area of π × radius × slant height, to which the base π × radius² is added. The slant height comes from the Pythagorean theorem on the right triangle formed by the radius, the vertical height, and the slant. The value of π used is the one tabulated by the US National Institute of Standards and Technology. For further reading, see Cone.

1. The Pythagorean theorem, giving the slant height ℓ = √(radius² + height²) from the radius and vertical height.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/