Sphere Surface Area Calculator

A sphere has no edges or corners, just one curved face, so a single length settles everything about it. Which length to use is a practical matter. You cannot reach the center of a ball to measure a radius, but a soft tape around its widest part takes seconds. A spherical tank on a drawing, on the other hand, is usually specified by its diameter.

Formula

A = 4 × π × r²

A

the surface area

π

pi, approximately 3.14159

r

the radius, from the center to the surface

Surface area from the radius

Square the radius, multiply by π, then multiply by 4. A sphere with a radius of 5 cm has a surface area of 4 × π × 25 ≈ 314.16 cm². Because the radius is squared, a sphere of twice the radius has four times the surface: a 10 cm radius gives about 1,256.64 cm².

Surface area from the diameter

The diameter is twice the radius, so the 4 cancels out and the formula shrinks to π times the diameter squared. A ball 24 cm across has a surface of π × 576 ≈ 1,809.56 cm². A spherical tank 8 ft wide has π × 64 ≈ 201.06 ft² of steel to paint, before allowing for legs, pipes or a second coat.

• A = π × d²

Surface area from the circumference

The circumference here means the great circle: the largest loop that fits around the sphere, centered on the same point, like the equator on a globe. Square it and divide by π. A ball measuring 70 cm around has a surface of 70² ÷ π ≈ 1,559.72 cm² and a radius of about 11.14 cm.

Keep the tape on the widest line. Any loop away from the middle is a smaller circle, which reads short and understates the area.

• A = C² ÷ π

Why the formula has a 4 in it

Slice a sphere through its center and the cut face is a flat circle with an area of πr². The curved surface is worth exactly four of those circles, a result Archimedes proved more than two thousand years ago. He also noticed a tidy way to see it: wrap a cylinder snugly around the sphere. Its height is 2r and its circumference is 2πr, so its curved wall is 2r × 2πr = 4πr², the same area as the sphere inside.

Surface area versus volume

Surface area, in square units, is how much paint, leather or foil covers the outside. Volume, in cubic units, is how much the sphere holds. The calculator shows both, and the volume calculator covers other solids. A sphere with a 5 cm radius holds 4 ÷ 3 × π × 125 ≈ 523.60 cm³.

The two grow at different rates. Doubling the radius multiplies the surface by 4 but the volume by 8, which is why a large tank has less skin per liter than a small one and loses heat more slowly for its size.

• V = 4 ÷ 3 × π × r³

Half a sphere

A hemisphere's curved part is half the sphere, 2πr². If the flat circular base counts as well, add πr² for a total of 3πr². A dome with a 3 m radius has a curved surface of 2 × π × 9 ≈ 56.55 m², and 3 × π × 9 ≈ 84.82 m² with its floor included.

Frequently asked questions