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Spheres: Definition, Area & Volume

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  • 0:05 Spheres
  • 0:44 Important Measurement
  • 1:02 Surface Area
  • 2:42 Volume
  • 4:09 Lesson Summary
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Lesson Transcript
Instructor
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

Watch this video lesson to learn how you can find the surface area and volume of spheres. Learn the formulas and measurements you need to solve problems and discover how to identify spheres in the real world.

Spheres

Spheres are three-dimensional shapes with no corners. Every point on the surface is the same distance from the center. We see spheres every day, and we don't think much about how they are defined.

Take a ball, for example. A ball is a sphere, but do we think about how it is defined? No, because most likely we will end up playing with it, bouncing it up and down. But take a close look, and you will see that it has no corners and that no matter how you hold the ball, every one of those points will be the same distance to the very center of the ball.

Important Measurement

This distance from any point on the sphere's surface to the center of the sphere is called the radius. Because a sphere is a uniform shape, meaning that no matter how you turn it, it will always look the same, the radius is the only measurement we need.

Surface Area

The formula to find a sphere's surface area, the area of just the surface of a three-dimensional object, requires the radius measurement.

Surface Area = 4 * pi * r2

The r stands for the radius, and the pi is approximated by 3.14.

Once we know the radius, we can plug this information into our formula and evaluate to find our surface area. Let's see how this works.

Let's say we have a basketball whose radius is 5 inches, and we want to find the surface area. What do we do?

Well, we first check to see if we know the radius. Yes, the problem tells us that the radius of our sphere, our basketball, is 5 inches. Oh good, we can now simply plug this number into our formula and find our answer. So we plug in 5 in for r.

Surface Area = 4 * 3.14 * 52

I've replaced pi with its approximation of 3.14 as well. Now I square the 5 to get 25 and then multiply that with the rest to get my answer.

Surface Area = 4 * 3.14 * 25

Surface Area = 314 inches squared

My answer is 314 inches squared. I recall that area is always squared so I have to make sure my answer has my measurement units squared. Everything checks out and I am done.

Volume

To find a sphere's volume, the amount of space inside a three-dimensional object, we need to know only the radius as well. And the process is similar to that of finding the surface area, just with a different formula.

Volume = (4/3) * pi * r3

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Additional Activities

Practice Problems - Surface Area and Volume of Spheres

In the following practice problems, students will calculate the volume and surface area for a sphere of a given radius, calculate the radius for a sphere of a given volume or surface area, and analyze how doubling the radius affects the volume and surface area of a sphere.

Practice Problems

1. Find the volume and surface area of a sphere with radius 3 inches. Use 3.14 for pi and round to the nearest hundredth.

2. A sphere has a volume of 40 cubic centimeters. What is the radius of the sphere? Use 3.14 for pi and round to the nearest hundredth.

3. A sphere has a surface area of 90 square inches. What is the radius of the sphere? Use 3.14 for pi and round to the nearest hundredth.

4. How does doubling the radius of a sphere affect the volume and surface area of the sphere? Are they doubled as well? If not, how do they change?

Solutions

1. Using the formulas for volume and surface area with r = 3 and pi = 3.14, we have


2. Using the formula for volume, with V = 40 and pi = 3.14, we can solve for r.

So the radius is about 2.12 centimeters.

3. Using the formula for surface area with A = 90 and pi = 3.14, we can solve for r.

So the radius is about 2.68 inches.

4. To find out what happens if the radius is doubled, replace r with 2r in the formulas for volume and surface area. For volume,

So the volume is not doubled - it is actually 8 times the original volume! For surface area,
The surface area is not doubled either - it is 4 times the original surface area.

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