Surface Area of Sphere

Formula for the surface area of spheres

Now suppose you are holding a perfectly spherical ball in your hand, and you want to tightly wrap it in paper. The surface area of the sphere can be thought of as the minimum amount of paper that would be required to completely cover its entire surface. In other words, the sphere's surface area is the space that covers the surface of the shape, measured with square units (i.e., m², ft², etc.)

Consider the following sphere with radius r:

Surface area of a sphere - StudySmarter Originals

The surface area, S, of the sphere with radius, r, is given by the following formula:

$S=4{\mathrm{\pi r}}^2$

Calculating the surface area of spheres with diameter

Suppose that instead of the radius, you are given the diameter of the sphere. Since the diameter is twice the length of the radius, we can simply substitute the value $r=d/2$ in the above formula. This would lead to:

$$\begin{gathered} S=4\pi r^2 \\ =4\pi {(d/2)}^2=4\pi (d²/4) \\ =\pi d² \end{gathered}$$

So the surface area, S, of a sphere with diameter, d, is:

$S={\mathrm{\pi d}}^2$

Great circles and the surface of spheres

When a plane intersects a sphere so that it contains the center of the sphere, the intersection is called a great circle. In effect, a great circle is a circle contained within the sphere whose radius is equal to the radius of the sphere. A great circle separates a sphere into two congruent halves, each called a hemisphere.

For example, if we approximate the Earth's shape as spherical, then we can call the equator a great circle because it passes through the center and splits the Earth (approximately) into two halves.

Examples using the surface area of sphere formula

Let us take a look at some examples related to surface area of spheres.