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# Areas of Trapezoids, Rhombuses, and Kites

Aug 21, 2023

## Properties of Trapezoid

• Only one pair of opposite sides is parallel to each other.
• The sum of adjacent angles on non-parallel sides is 180°.

## Properties of Rhombus

• All sides of the rhombus are equal.
• The opposite sides of a rhombus are parallel.
• Opposite angles of a rhombus are equal.
• In a rhombus, diagonals bisect each other at right angles.
• Diagonals bisect the angles of a rhombus.
• The sum of two adjacent angles is equal to 180 degrees.

## Properties of Kite

• The pair of adjacent sides are equal.
• One pair of opposite angles is equal.
• Diagonals are perpendicular to each other
• The longer diagonal bisects the shorter diagonal.

### Area of Trapezoid

The bases of a trapezoid are the parallel lines and the height of a trapezoid is the perpendicular distance between the two bases.

The area of a trapezoid is one-half the product of the height and the sum of the lengths of the bases. (b1 and b2 are the length of the bases and h is the perpendicular distance between the two bases)

In a trapezoid, the average of the lengths of the bases is also the length of the midsegment. So, you can also find the area by multiplying the midsegment by the height.

### Area of Rhombus

Method 1:

The area of a rhombus is one-half the product of the lengths of its diagonals.

Length of diagonals: d1 and d2

Method 2:

Rhombus is also a parallelogram.

If the length of base (b) and height (h) are given, then we can simply use the formula of the area of a parallelogram.

A = b × h

### Area of Kite

The area of a kite is one-half the product of the lengths of its diagonals.

The lengths of the diagonals are d1 and d2.

#### Let’s Solve Some Examples!

Example: 1

The free-throw lane on an international basketball court is shaped like a trapezoid. Find the area of the free-throw lane.

Solution:

The height of the trapezoid is 5.8 m. The lengths of the bases are 3.6 m and 6 m.

b1 = 3.6 m, b2 = 6 m, h = 5.8 m

A =  1/2 × h × (b1+b2)

=  1/2 × 5.8 × (3.6 + 6)

= 27.84

The area of the free-throw lane is 27.84 square meters.

#### Example: 2

Rhombus PQRS represents one of the inlays on the guitar in the photo. Find the area of the inlay.

Solution:

1. QN = 9 mm, PN = 12mm

The diagonals of a rhombus bisect each other so:

QN = NS, PN = NR

The lengths of the diagonals are:

QS (d1) = 18 mm, PR (d2) = 24 mm

• Area of a rhombus:

A =  1/2 × d1 × d2

=  1/2 × 18 × 24 = 216

The area of the inlay is 216 square mm.

#### Example: 3

One diagonal of a kite is twice as long as the other diagonal. The area of the kite is 72.25 square inches. What are the lengths of the diagonals?

Solution:

Let x be the length of one diagonal. Then 2x is the length of the other diagonal.

Area of kite:

A = 1/2 × d1 × d2

A = 1/2 × x × 2x

72.25 =  1/2 × 2 × x2

x2 = 72.25

x = 8.5

The lengths of the diagonals are d1 = 8.5 in and d2 = 17 in.

#### Questions to Solve

Question: 1

The lengths of the bases of a trapezoid are 5.4 centimeters and 10.2 centimeters. The height is 8 centimeters. Draw and label a trapezoid that matches this description. Then find its area.

Solution:

The lengths of the bases are b1=5.4 cm,

b2= 10.2 cm

Height (h) = 8 cm

Area = 1/2 × h × (b1 + b2) =  1/2 × 8 × (5.4 + 10.2) = 62.4 square cm.

Question: 2

Find the area of the below figure. Tell whether it is a rhombus or a kite.

Solution:

It is a kite because the longer diagonal is divided into two unequal parts.

Longer diagonal: d1 = 4 + 5 = 9 unites

Shorter diagonal: d2 = 2 + 2 = 4 units (the longer diagonal bisects the shorter diagonal in a kite)

A =  × d1 × d2 =  × 9 × 4 = 18 square units

Question: 3

The figure is a rhombus. Its side length is 13. The length of one of its diagonals is 24. Sketch the figure and determine its area.

Solution:

There is a right-angled triangle formed (ABC).

Sides of the triangle: AC =13, AB = 12, BC = x

Following the Pythagoras theorem,

122 + x2 = 132

x = 5 units

Lengths of the diagonals are: d1 = 24, d2 = 10

A = 1/2 × d1 × d2 = 1/2 × 24 × 10 = 120 square units

#### Exercise:

1. Find the area of a trapezoid with bases of 12 cm and 18 cm and a height of 10 cm.
2. Find the area of a trapezoid with bases 2 ft and 3 ft and height 1/3 ft.
3. The border of Tennessee resembles a trapezoid with bases of 342 mi and 438 mi and a height of 111 mi. Approximate the area of Tennessee by finding the area of this trapezoid.
4. The area of a kite is 120 square cm. The length of one diagonal is 20 cm. What is the length of the other diagonal?
5. The lengths of the sides of a rhombus and one of its diagonals are each 10 m. What is the area of the rhombus?
6. The area of a trapezoid is 100 square ft. The sum of the two bases is 25 ft. What is the height of the trapezoid?
7. Find the area of a trapezoid with bases of 3 cm and 19 cm and a height of 9 cm.
8. Find the area of a trapezoid in a coordinate plane with vertices at (1, 1), (1, 6),
(5, 9), and (5, 1).
9. One base of a trapezoid is twice the other. The height is the average of the two bases. The area is 324 square cm. Find the height and the bases. (Hint: Let the smaller base be 2x.)
10. Find the area of the rhombus with diagonals 10 cm and 20 cm.

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