## Finding the Volume and Surface Area of Spheres

Contents

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 \(\pi \) with \(3.14.\)

### Definition: Volume and Surface Area of a Sphere

For a sphere with radius \(r\text{:}\)

## Example

A sphere has a radius \(6\) inches. Find its volume and surface area.

### Solution

Step 1 is the same for both and , so we will show it just once.

Step 1. Read the problem. Draw the figure and labelit with the given information. |

Step 2. Identify what you are looking for. | the volume of the sphere |

Step 3. Name. Choose a variable to represent it. | let V = volume |

Step 4. Translate.Write the appropriate formula. | \(V=\frac{4}{3}\pi {r}^{3}\) |

Step 5. Solve. | \(V\approx \frac{4}{3}\left(3.14\right){6}^{3}\) \(V\approx 904.32\phantom{\rule{0.2em}{0ex}}\text{cubic inches}\) |

Step 6. Check: Double-check your math on a calculator. | |

Step 7. Answer the question. | The volume is approximately 904.32 cubic inches. |

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. | \(S=4\pi {r}^{2}\) |

Step 5. Solve. | \(S\approx 4\left(3.14\right){6}^{2}\) \(S\approx 452.16\) |

Step 6. Check: Double-check your math on a calculator | |

Step 7. Answer the question. | The surface area is approximately 452.16 square inches. |

## Example

A globe of Earth is in the shape of a sphere with radius \(14\) centimeters. Find its volume and surface area. Round the answer to the nearest hundredth.

### Solution

Step 1. Read the problem. Draw a figure with thegiven information and label it. |

Step 2. Identify what you are looking for. | the volume of the sphere |

Step 3. Name. Choose a variable to represent it. | let V = volume |

Step 4. Translate.Write the appropriate formula. Substitute. (Use 3.14 for \(\pi \)) | \(V=\frac{4}{3}\pi {r}^{3}\) \(V\approx \frac{4}{3}\left(3.14\right){14}^{3}\) |

Step 5. Solve. | \(V\approx 11,488.21\) |

Step 6. Check: We leave it to you to check your calculations. | |

Step 7. Answer the question. | The volume is approximately 11,488.21 cubic inches. |

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 \(\pi \)) | \(S=4\pi {r}^{2}\) \(S\approx 4\left(3.14\right){14}^{2}\) |

Step 5. Solve. | \(S\approx 2461.76\) |

Step 6. Check: We leave it to you to check your calculations. | |

Step 7. Answer the question. | The surface area is approximately 2461.76 square inches. |