# ISEE Upper Level Math : Trapezoids

## Example Questions

### Example Question #1 : Trapezoids

The bigger base length of a trapezoid is two times bigger than the smaller base length. Legs of the trapezoid have the same length of . If the perimeter of the trapezoid is , give the length of the smaller base.

Explanation:

Like any polygon, the perimeter of a trapezoid is the total distance around the outside, i.e.

,

where are the lengths of each side.

Let small base length and big base length.

### Example Question #2 : Trapezoids

The perimeter of the following trapezoid is .

The following is given:

Find .

Figure not drawn to scale.

Explanation:

The perimeter of a trapezoid is equal to the sum of all of the sides, i.e.

,

where are the lengths of each side.

### Example Question #1 : How To Find The Area Of A Trapezoid

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is  ?

Explanation:

Substitute  in the formula for the area of a trapezoid:

### Example Question #1 : How To Find The Area Of A Trapezoid

A trapezoid has height 20 inches. Its bases have sum 30 inches, and one base is 6 inches longer than the other. What is the area of this trapezoid?

It cannot be determined from the information given.

Explanation:

You do not need to find the individual bases; their sum, , is . You can substitute  into the formula for the area of a trapezoid:

square inches.

Note that the fact that one base is  inches longer is not important here.

### Example Question #5 : Trapezoids

Find the area of the above trapezoid if , , and .

Figure not drawn to scale.

Explanation:

The area of a trapezoid is given by

,

where  are the lengths of each base and is the altitude (height) of the trapezoid.

### Example Question #6 : Trapezoids

A trapezoid has the base lengths of and . The area of the trapezoid is . Give the height of the trapezoid in terms of .

Explanation:

The area of a trapezoid is given by

,

where  are the lengths of each base and is the altitude (height) of the trapezoid.

### Example Question #7 : Trapezoids

In the following trapezoid  and . The area of the trapezoid is 54 square inches. Give the height of the trapezoid. Figure not drawn to scale.

Explanation:

The area of a trapezoid is given by

,

where  are the lengths of each base and is the altitude (height) of the trapezoid.

Substitute these values into the area formula:

### Example Question #8 : Trapezoids

What is the area of the shaded portion of the above square?

Explanation:

Quadrilateral  - the shaded region - is a trapezoid with bases  and , and altitude . The area of the trapezoid can be calculated using the formula

,

where  and  , and .

The length of  can be found by setting  and  and applying the Pythagorean Theorem:

Therefore,

.

Substituting:

### Example Question #9 : Trapezoids

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?