Properties of Trapezoid
![Trapezoid](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Trapezoid.png)
- Only one pair of opposite sides is parallel to each other.
- The sum of adjacent angles on non-parallel sides is 180°.
Trapezoid Shapes in Real Life:
![Aeroplane](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Aeroplane.png)
![Popcorn](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Popcorn.png)
![Bridge](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Bridge.png)
Properties of Rhombus
![Rhombus](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Rhombus.png)
- 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.
Rhombus Shapes in Real Life:
![Grid](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Grid.jpg)
![Kites](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Kites.jpg)
Properties of Kite
![Properties of Kite](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Properties-of-Kite.png)
- 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.
Kite Shapes in Real Life:
![Kites](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Kites-2.jpg)
![Drop](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Drop.jpg)
![Diagonal-Board](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Diagonal-Board.jpg)
How to calculate the Area of the Trapezoid, Rhombus, and Kite?
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.
![Area of Trapezoid](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Area-of-Trapezoid.png)
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.
![trapezoid](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Trapezoid-1.png)
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
![area of a rhombus](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Area-of-a-rhombus.png)
![parallelogram](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/parallelogram.png)
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.
![area of a kite](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/area-of-a-kite.png)
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.
![trapezoid](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Trapezoid-2.png)
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.
![guitar](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/guitar.png)
![Rhombus PQRS](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Rhombus-PQRS.png)
Solution:
- 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:
![The length of one diagonal](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/The-length-of-one-diagonal.png)
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
![trapezoid are 5.4](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/trapezoid-are-5.4.png)
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.
![a rhombus or kite](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/a-rhombus-or-kite.png)
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:
![diagonals is 24](https://www.turito.com/learn-internal/wp-content/uploads/2023/08/Diagonals-is-24.png)
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:
- Find the area of a trapezoid with bases of 12 cm and 18 cm and a height of 10 cm.
- Find the area of a trapezoid with bases 2 ft and 3 ft and height 1/3 ft.
- 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.
- 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?
- The lengths of the sides of a rhombus and one of its diagonals are each 10 m. What is the area of the rhombus?
- 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?
- Find the area of a trapezoid with bases of 3 cm and 19 cm and a height of 9 cm.
- Find the area of a trapezoid in a coordinate plane with vertices at (1, 1), (1, 6),
(5, 9), and (5, 1). - 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.)
- Find the area of the rhombus with diagonals 10 cm and 20 cm.
![Area of Trapezoids, rhombuses and kites](/_next/image?url=https%3A%2F%2Fwww.turito.com%2Flearn-internal%2Fwp-content%2Fuploads%2F2023%2F08%2FArea-of-Trapezoids-rhombuses-and-kites.png&w=1920&q=50)
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