Surface Area and Volume of a Sphere

In mathematics, a sphere is a three-dimensional geometric figure in which the distance between the surface and the center is equal for every point on the surface. A sphere is the solid figure obtained by rotating a circle around its diameter.

You can compute the volume and surface area of a sphere if you know the length of its radius, i.e., the distance between the center and the outside of the sphere. You can also work backwards and compute the radius from the surface area, or the radius from the volume. The sphere calculator below will tell you all three measures of a sphere (radius, surface area, volume) if you input one of the values. Or you can use the geometric formulas below.

Spherical Volume Formula

For a sphere with a radius of R, the formula for its volume is

Volume = (4/3)*pi*R³,

where pi is the constant 3.14159265358...

Example 1: Suppose a spherical beach ball has a diameter of 14 inches. Its radius is then 7 inches, and so its volume is

(4/3)(3.14159)(7³) = 1436.755 cubic inches.

Spherical Surface Area Formula

For a sphere with a radius of R, the formula for its surface area is

Surface Area = 4*pi*R².

Notice that the expression for surface area is the derivative of the volume equation with respect to R. This is no coincidence! In two dimensions, the area of a circle is pi*R². The circumference is 2*pi*R, which is the derivative of the expression for area.

Example 2: Suppose a spherical globe has a diameter of 20 cm. Its radius is then 10 cm, and thus its surface area is

(4)(3.14159)(10²) = 1256.637 square cm.

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