Surface Area Of Sphere Calculator

What this calculator does

The surface area of a sphere is the whole of its curved outside, the area you would need to cover a perfectly round ball. Because a sphere is the same in every direction, one number settles it: the radius.

Type the radius and you have the area of the outside.

Using the calculator

1. Type the radius, from the centre to the surface.
2. Press Calculate.

The radius has to be positive. There is no unit setting, so the result is a plain number in square units of whatever unit you used. If you only have the diameter, the distance straight across, halve it first.

The formula | surface area = 4 × π × radius²

The surface area of a sphere is:

surface area = 4 × π × radius²

Short, and a little surprising, because it is such a clean multiple of the area of a circle. That 4 is the interesting part, and it is no accident.

Four great circles

A sphere's surface area is exactly four times the area of its great circle, the flat circle of radius r you would get by slicing the sphere through its centre, whose area is π × radius². So 4 × π × radius² is just four of those circles. A piece of string that could cover the whole sphere would cover exactly four such circles laid flat. Archimedes proved this more than two thousand years ago, and counted it among his finest results.

The sphere that will not lie flat

A cone or a cylinder can be cut and unrolled flat, into a sector or a rectangle, and its surface area read straight off. A sphere cannot. There is no way to flatten a sphere's surface without stretching or tearing it, which is exactly why every flat map of the round Earth has to distort something, whether shapes, sizes or distances.

So Archimedes found the area another way: a sphere has the same surface area as the curved side of the cylinder that just contains it, the cylinder of radius r and height 2r. That side is 2 × π × radius × 2 × radius, which is 4 × π × radius². The same relationship is what makes equal-area cylindrical world maps possible. See the surface area of a cylinder calculator for that side area.

Double the radius, four times the area

Because the radius is squared, a sphere's surface area climbs quickly. Double the radius and the area does not double, it goes up four times, since 2 squared is 4. A ball twice as wide takes four times as much to cover.

Square units, and the π it uses

Surface area is an area, so the answer is in square units of whatever you measured in: a sphere given in centimetres gives cm². The tool shows a plain number with no unit attached. For π it uses 3.141592654, ten significant figures, finer than any radius you could measure.

A worked example | radius 10

Say the radius is 10.

1. Square the radius: 10² = 100.
2. Multiply by 4 and by π: 4 × π × 100 ≈ 1,256.64.

So the surface area is about 1,256.64 square units. As a check on the four-circles idea, one great circle here is π × 10² ≈ 314.16, and four of those is the same 1,256.64.

Questions people ask

What is the surface area of a sphere with radius 10?

About 1,256.64 square units. Square the radius, then multiply by 4 and by π.

Why four?

Because a sphere's surface area is exactly four times the area of its great circle, π × radius², a result proved by Archimedes.

Do I use the radius or the diameter?

The radius. If you have the diameter, the distance straight across, halve it first.

Can a sphere be unrolled flat like a cylinder?

No. A sphere's surface cannot be flattened without distortion, which is why flat world maps always stretch something. Its area is found indirectly, via the cylinder that contains it.

What happens to the area if I double the radius?

It grows four times, not two, because the radius is squared and 2 squared is 4.

References

A note on where this comes from. That a sphere's surface area is four times the area of its great circle, equal to the curved side of its circumscribing cylinder, was proved by Archimedes in On the Sphere and Cylinder, the same work that gives the sphere's volume. This equal-area relationship underlies the Lambert cylindrical map projection. The value of π used is the one tabulated by the US National Institute of Standards and Technology. For further reading, see Sphere.

1. Archimedes, On the Sphere and Cylinder (c. 225 BCE), showing a sphere's surface area is four times its great circle, equal to the side of its circumscribing cylinder.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π. https://dlmf.nist.gov/