Learning Outcomes
- Find the volume and surface area of spheres
A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.
Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate [latex]\pi [/latex] with [latex]3.14[/latex].
Volume and Surface Area of a Sphere
For a sphere with radius [latex]r\text{:}[/latex]
example
A sphere has a radius [latex]6[/latex] inches. Find 1. its volume and 2. its surface area.
Solution
Step 1 is the same for both 1. and 2., so we will show it just once.
Step 1. Read the problem. Draw the figure and label
it with the given information. |
1. | |
Step 2. Identify what you are looking for. | The volume of the sphere |
Step 3. Name. Choose a variable to represent it. | Let [latex]V[/latex] = volume. |
Step 4. Translate.
Write the appropriate formula. |
[latex]V=\Large\frac{4}{3}\normalsize\pi {r}^{3}[/latex] |
Step 5. Solve. | [latex]V\approx \Large\frac{4}{3}\normalsize\left(3.14\right)({6}in)^{3}[/latex]
[latex]V\approx 904.32in^3[/latex] |
Step 6. Check: Double-check your math on a calculator. | |
Step 7. Answer the question. | The volume is approximately [latex]904.32[/latex] cubic inches. |
2. | |
Step 2. Identify what you are looking for. | The surface area of the cube |
Step 3. Name. Choose a variable to represent it. | Let S = surface area. |
Step 4. Translate.
Write the appropriate formula. |
[latex]S=4\pi {r}^{2}[/latex] |
Step 5. Solve. | [latex]S\approx 4\left(3.14\right)({6}in)^{2}[/latex]
[latex]S\approx 452.16in^2[/latex] |
Step 6. Check: Double-check your math on a calculator | |
Step 7. Answer the question. | The surface area is approximately [latex]452.16[/latex] square inches. |
try it
example
A globe of Earth is in the shape of a sphere with radius [latex]14[/latex] centimeters. Find 1. its volume and 2. its surface area. Round the answer to the nearest hundredth.
try it
In the next video we show an example of how to find the surface area of a sphere.
And in our final video we show an example of how to find the volume of a sphere given it’s diameter.