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Study Guides > ALGEBRA / TRIG I

Finding the Volume and Surface Area of a Sphere

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] An image of a sphere is shown. The radius is labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.
 

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

[ohm_question]146804[/ohm_question]
 

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.

Answer: Solution

Step 1. Read the problem. Draw a figure with the given information and label it. .
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. Substitute. (Use 3.14 for [latex]\pi [/latex] ) [latex]V=\Large\frac{4}{3}\normalsize\pi {r}^{3}[/latex] [latex]V\approx \Large\frac{4}{3}\normalsize\left(3.14\right)({14}cm)^{3}[/latex]
Step 5. Solve. [latex]V\approx 11,488.21cm^3[/latex]
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The volume is approximately [latex]11,488.21[/latex] cubic centimeters.
2.
Step 2. Identify what you are looking for. The surface area of the sphere
Step 3. Name. Choose a variable to represent it. Let S = surface area.
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for [latex]\pi [/latex] ) [latex]S=4\pi {r}^{2}[/latex] [latex]S\approx 4\left(3.14\right)({14}cm)^{2}[/latex]
Step 5. Solve. [latex]S\approx 2461.76cm^2[/latex]
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The surface area is approximately [latex]2461.76[/latex] square centimeters.

 

try it

[ohm_question]146806[/ohm_question]
In the next video we show an example of how to find the surface area of a sphere. https://youtu.be/OLlEbk1xGm4 And in our final video we show an example of how to find the volume of a sphere given it's diameter. https://youtu.be/_kejcXbRjGY

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