First, let's visualize the components of a sphere. Consider congruent circles in three-dimensional space that all have the same point for their center. Taken together, these circles form a sphere. All points on the surface of the sphere are an equal distance from its center. This distance is the **radius** of the sphere.

In space, a sphere is the locus of all points that are at a given distance from a given point called its center.

## 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., m^{2}, ft^{2}, etc.)

Consider the following sphere with radius r:

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:

$S=4\pi {r}^{2}\phantom{\rule{0ex}{0ex}}=4\pi {(d/2)}^{2}=4\pi (d\xb2/4)\phantom{\rule{0ex}{0ex}}=\pi d\xb2$

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.

Find the surface area of a sphere of radius 5 ft.

Solution:

$S=4\pi {r}^{2}\phantom{\rule{0ex}{0ex}}=4\times \mathrm{\pi}\times {5}^{2}\phantom{\rule{0ex}{0ex}}=314.29f{t}^{2}$

Find the surface area of a sphere given that the area of its great circle is 35 square units.

Solution:

Surface area of the sphere = 4πr^{2}

Area of the great circle = πr^{2}

We are given

πr^{2} = 35

Surface area of the sphere = 4πr^{2}

= 4 × 35

= 140 square units

The surface area of a sphere is 616 ft^{2}. Find its radius.

Solution:

$S=4\pi {r}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow 616=4\times \mathrm{\pi}\times {r}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {r}^{2}=\frac{616}{4\times \mathrm{\pi}}\phantom{\rule{0ex}{0ex}}\Rightarrow r=\sqrt{49}\phantom{\rule{0ex}{0ex}}=7$

Note: The radius must be a positive value, so we know that -7 is not the solution.

## Surface of spheres - Key takeaways

- In space, a sphere is the locus of all points that are at a given distance from a given point called its center.
- The surface area, S, of a sphere with radius, r, is given by the formula:S = 4πr²
- When a plane intersects a sphere so that it contains the center of the sphere, the intersection is called a great circle.

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##### Frequently Asked Questions about Surface Area of Sphere

What is the surface area of spheres?

The sphere's surface area is the space that covers the surface of the shape, measured with square units (m^{2}, ft^{2}, etc.). The surface area, S, of a sphere with radius, r, is given by the formula: S = 4πr^{2}

What is the formula for calculating the surface area of sphere?

The surface area, S, of a sphere with radius, r, is given by the formula: S = 4πr^{2}

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