## Using the Properties of Trapezoids

Contents

A **trapezoid** is four-sided figure, a *quadrilateral*, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base \(b,\) and the length of the bigger base \(B.\) The height, \(h,\) of a trapezoid is the distance between the two bases as shown in the figure below.

The formula for the **area of a trapezoid** is:

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See the figure below.

The height of the trapezoid is also the height of each of the two triangles. See the figure below.

The formula for the area of a trapezoid is

If we distribute, we get,

### Definition: Properties of Trapezoids

- A trapezoid has four sides as shown above.
- Two of its sides are parallel and two sides are not.
- The area, \(A,\) of a trapezoid is \(\text{A}=\frac{1}{2}h\left(b+B\right)\).

## Example

Find the area of a trapezoid whose height is 6 inches and whose bases are \(14\) and \(11\) inches.

### Solution

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

Step 2. Identify what you are looking for. | the area of the trapezoid |

Step 3. Name. Choose a variable to represent it. | Let \(A=\text{the area}\) |

Step 4.Translate.Write the appropriate formula. Substitute. | |

Step 5. Solve the equation. | |

Step 6. Check: Is this answer reasonable? |

If we draw a rectangle around the trapezoid that has the same big base \(B\) and a height \(h,\) its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base \(b\) and a height \(h,\) its area should be smaller than that of the trapezoid.

The area of the larger rectangle is \(84\) square inches and the area of the smaller rectangle is \(66\) square inches. So it makes sense that the area of the trapezoid is between \(84\) and \(66\) square inches

Step 7. **Answer** the question. The area of the trapezoid is \(75\) square inches.

## Example

Find the area of a trapezoid whose height is \(5\) feet and whose bases are \(10.3\) and \(13.7\) feet.

### Solution

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

Step 2. Identify what you are looking for. | the area of the trapezoid |

Step 3. Name. Choose a variable to represent it. | Let A = the area |

Step 4.Translate.Write the appropriate formula. Substitute. | |

Step 5. Solve the equation. | |

Step 6. Check: Is this answer reasonable?The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5. | |

Step 7. Answer the question. | The area of the trapezoid is 60 square feet. |

## Example

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of \(3.4\) yards and the bases are \(8.2\) and \(5.6\) yards. How many square yards will be available to plant?

### Solution

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

Step 2. Identify what you are looking for. | the area of a trapezoid |

Step 3. Name. Choose a variable to represent it. | Let A = the area |

Step 4.Translate.Write the appropriate formula. Substitute. | |

Step 5. Solve the equation. | |

Step 6. Check: Is this answer reasonable?Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd. | |

Step 7. Answer the question. | Vinny has 23.46 square yards in which he can plant. |

### Optional Video: Area of a Trapezoid

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