Spheres

Introduction: We examine the 3-dimensional object known as a sphere.

Definitions:

A sphere is a 3-dimensional solid which comprises all points in space that are equi-distant from one point called the center of the sphere. The distance (sometimes the line segment itself) is called the radius of the sphere. We show sphere P below. It is named P because P is the center of the sphere. The radius r is the segment (distance) PQ.

The Lesson:

Two important formulas help us measure the size of a sphere.

• volume =
• surface area =

In the sphere shown above, suppose that r = PQ = 5 inches in length. The volume would be given by

.

While the surface area would be given by

.

Let's Practice:

1. If a sphere has radius 9 meters, what are the surface area and the volume?

To find surface area, we use

.

To find the volume, we use

.

2. Suppose a sphere has a surface area of . What is the volume?

We have surface area = . We solve for r by dividing both sides of this equation by . This gives us

.

To calculate the volume we use

.