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# Area of a Trapezoid

A trapezoid is a quadrilateral that has only one pair of sides that are parallel. To calculate the area of trapezoids, take the average of bases and multiply by its height.

The area of a trapezoid is given by:

A = (a + b) ∙ h /2

a: the length of the top
b: the length of the base
h: the height

When a = 0 the shape becomes a triangle.

Example 1:
What is the area of a trapezoid having bases 5cm and 8cm and a height of 6cm?
Solution:
Using the formula for the area of a trapezoid, we will get:

Area of Trapezoid = 0.5 × h × (a + b)
Area of Trapezoid = 0.5 × 6 × (5 + 8)
Area of Trapezoid = 0.5 × 6 × 13

Area of Trapezoid = 39 cm2

Example 2:
The area of a trapezoid is 52 cm2 and the bases are 11 inches and 15 inches respectively. Find its height.
Solution: We know that the area of a trapezoid is given by:
Area of Trapezoid = 0.5× h × (a + b), where h is the height, by isolating h from the formula we will able to determine its height
Area = 0.5× h × (a + b)
or
52 = 0.5 × (11 + 15) × h
52 = 0.5 × 26 × h
52 = 13h
Thus:
h = 52/13 = 4 inches

Example 3: Area of a trapezoid is 15 cm2 and the distance between the parallel bases is 6 cm. If one of the parallel bases is 3 cm, then what is the length of the other parallel base?
Solution: Let a be the length of the unknown parallel side and b be the known base.

Area of the trapezoid = 0.5 × height × (a+b) = 15 cm2
Substituting the values we will get:
(0.5) × 6 × (3 + a) = 15

Multiply each side by 2
6 x (3 + a) = 30
Dividing each side by 6, we will get,
3 + a = 5
a = 2 cm
Hence, the length of the other parallel side is 2 cm.

Example 4:  What are the lengths of the parallel sides of a trapezoid, if its area is 18 cm2, height is 4 cm and the length of its shorter side is 5 cm shorter than its longer side?
Solution: Let y be the length of the longer side.

The length of the shorter side is (y - 5) cm, since shorter side is 5 cm shorter than the longer side.

Area of the trapezoid = 18 cm2
According to the formula for the area of a trapezoid we have:
(0.5) × 4 × [y + (y - 5)] = 18

Multiply each side by 2,
4 × (2y - 5) = 36
Divide each side by 4,
2y - 5 = 9
Simplify the equation we will get:
2y = 14 and y = 7 cm

The length of the longer side is therefore y = 7 cm, whereas the length of the shorter side is y - 5 = 7 - 5 = 2 cm

Example 5: Area of a trapezium is 160 cm2. The parallel sides are 18 cm and 14 cm. Find the distance between the parallel sides.
Solution:
Given A = 160 cm2, a = 18 cm and b = 14 cm.
Area of the trapezium, A = (0.5) × (a + b) × h
Inserting the known values we have:

160 = (0.5) × (18 + 14) × h
i.e. (0.5) × (18 + 14) × h = 160
0.5
× 32 × h = 160
16 × h = 160

Divide by 16 each side,
h = 10 cm
The distance between the parallel sides is therefore 10 centimeter.

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