10.6 Area of Trapezoids, Rhombuses, and Kites

Learning Objectives

I can find the area and perimeter of a trapezoid given its two bases and its height.
I can find the area and perimeter of a kite or a rhombus given its two diagonals.

Introduction

What if you were given a trapezoid and the size of its two bases as well as its height? How could you find the total distance around the trapezoid and the amount of space it takes up? What if you were given a kite or a rhombus and the size of its two diagonals? How could you find the total distance around the kite or rhombus and the amount of space it takes up?

After completing this Concept, you'll be able to use the formulas for the perimeter and area of trapezoids and kites or rhombuses to solve problems like this.

Guided Learning

Area and Perimeter of Trapezoids

A trapezoid is a quadrilateral with one pair of parallel sides. The parallel sides are called the bases and we will refer to the lengths of the bases as b_1 and b_2 . The perpendicular distance between the parallel sides is the height of the trapezoid.

The area of a trapezoid is $A=\frac{1}{2} h(b_1+b_2)$ where is always perpendicular to the bases.

Take a moment to write down this formula in your journal.

Area of a trapezoid: $A=\frac{1}{2} h(b_1+b_2)$

Example A

Find the area of the trapezoid below.

$A = \frac{1}{2}h\left ( b_1 + b_2\right )$

$A & = \frac{1}{2} (11)(14+8)\\A & = \frac{1}{2} (11)(22)\\A & = 121 \ units^2$

Example B

Find the area of the trapezoid below.

$A & = \frac{1}{2} (9)(15+23)\\A & = \frac{1}{2} (9)(38)\\A & = 171 \ units^2$

Example C

Find the perimeter and area of the trapezoid. Remember the perimeter is the total length of all the sides.

P = 6 + 8 + 6+ 15 = 35 units

A = 1/2 h (b1 + b2)

A = 1/2 (4)(8 + 15)

A = 46 units2

Guided Practice

Find the perimeter and area of the following shapes. Round your answers to the nearest hundredth.

Stop and check your work with a partner.

Solutions

Use the formula for the area of a trapezoid.

1. P = 21 + 24 + 41 + 22 = 108 units

A = 1/2 h (b1 + b2) = 1/2 (18)(41 + 21) = 558 units2

2. P = 8 + 10 + 14 + 9 = 41 units

A = 1/2 h (b1 + b2) = 1/2 (7)(14 + 8) = 77 units2

3. P = 9 + 9 + 16 + 6 = 40 units

A = 1/2 h (b1 + b2) = 1/2 (5)(16 + 9) = 62.5 units2

Area and Perimeter of Rhombuses and Kites

Recall that a rhombus is a quadrilateral with four congruent sides and a kite is a quadrilateral with distinct adjacent congruent sides. Both of these quadrilaterals have perpendicular diagonals, which is how we are going to find their areas.

Notice that the diagonals divide each quadrilateral into four triangles. If we move the two triangles on the bottom of each quadrilateral so that they match up with the triangles above the horizontal diagonal, we would have two rectangles.

So, the height of these rectangles is half of one of the diagonals and the base is the length of the other diagonal. A formula for finding the area of a rhombus or kite has been created using that information.

The area of a rhombus or a kite is found by using the formula $A=\frac{1}{2} d_1 d_2$ .

Take a moment to write down this formula in your journal.

Area of a rhombus or kite: $A=\frac{1}{2} d_1 d_2$

Example D

Find the perimeter and area of the rhombus below.

In all rhombuses like this one, all sides are equal. The sides are labeled as equal by having the same markings on each side.

P = 4 x 14 = 56 units.

$A=\frac{1}{2} d_1 d_2$ = 1/2 (24)(16) = 192 units2

Example E

Find the perimeter and area of the kite below.

Notice that there are two different side lengths on this kite. The sides that are equal are marked in the same manner. The sides equal to 8 units are marked with one line. The sides equal to 15 units are marked with two lines. Look for markings like these to determine equal sides in geometric shapes.

P = 8 + 15 + 15 + 8 = 46 units

$A=\frac{1}{2} d_1 d_2$ = 1/2 (18)(9) = 81 units2

Example F

Find the perimeter and area of the rhombus below.

P = 4 x 16 = 64 units

The formula for area of rhombuses or kites is $A=\frac{1}{2} d_1 d_2$ . This figure is only showing the measurements for 1/2 of the length of the diagonals. In order to calculate the area, you will need to find the length of the entire diagonal.

$A=\frac{1}{2} d_1 d_2$ = 1/2 (28)(18)

= 252 units2

Guided Practice

Find the perimeter and area of the kites below.

3. Find the area of a rhombus with diagonals of 6 inches and 8 inches.

Stop and check your answers with a partner.

Solutions

1. Notice that only two sides of the kite are labeled. The equal sides are marked with lines.

P = 25 + 35 + 25 + 35 = 120 units

$A=\frac{1}{2} d_1 d_2$ = 1/2 (52)(40) = 1040 units2

2. It is important to note that the entire length of the diagonals are not labeled. You need to calculate the length of the diagonals before you can find the area.

P = 8 + 13 + 8 + 13 = 42 units

$A=\frac{1}{2} d_1 d_2$ = 1/2 (18)(10) = 90 units2

3. The area is $A=\frac{1}{2} d_1 d_2$ = 1/2 (8)(6) = 24 in2.

Review

To find the area of a trapezoid use the formula $A=\frac{1}{2} h(b_1+b_2)$ .
Perimeter is the total length of all the sides. Add all four sides of quadrilaterals to find the perimeter.
To find the area of a rhombus or kite use the formula $A=\frac{1}{2} d_1 d_2$ .

Perimeter

Perimeter is the distance around the edge of a figure. The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write “units.”

Area

Area is the space inside the edges of a figure. Area is measured in square units.

Trapezoid

A quadrilateral with one pair of parallel sides is called a trapezoid.

Rhombus

A rhombus is a quadrilateral with four congruent sides, opposite parallel sides, and opposite equal angles.

Kite

A quadrilateral with distinct adjacent congruent sides is called a kite.

Additional Resources

Area of a Trapezoid Video

Area of a Kite Video