What is a Trapezoid?
A trapezoid is a 2D closed figure having four sides with two opposite sides being parallel and the other two opposite sides being non-parallel. The parallel sides are called bases and the non-parallel sides are called legs.
The perpendicular distance between the parallel sides or the bases is called the height of the trapezoid.
In the image we can observe a shape that represents a trapezoid, ABCD, with bases or parallel sides AB and CD, that implies \(AB \parallel ~CD \) and legs BC and AD, which are the non parallel sides. The perpendicular distance between the bases is the height of the trapezoid, h.
[Note: BC and AD are not parallel to each other.]
How to find the Area of a Trapezoid?
Like any other 2D shape, the area of a trapezoid is the measure of the space or region enclosed within its four sides. We can calculate the area of a trapezoid using the following formula:
Area of trapezoid, A = \(\frac{1}{2} \) × (Sum of bases) × height
A = \(\frac{1}{2} ~\times~(b_1~+~b_2)\times~h\)
where, \(b_1\) and \(b_1\) are the bases and h is the height of the trapezoid.
So, the area of a trapezoid is one-half the product of its height, h and sum of its bases \(b_1\) and \(b_2\). Since, the bases and height are measured in units of length, the area is measured in square units.
What are the Properties of a Trapezoid?
- A trapezoid is a type of quadrilateral, so it has four sides.
- The sum of all its internal angles is 360°.
- Trapezoids have one pair of parallel sides called bases and one pair of non-parallel sides called legs.
Solved Examples
Example 1: Find the area of the trapezoid.
Solution:
As per figure:
\(b_1\) = 8 mm
\(b_2\) = 4 mm
h = 5 mm
\(A=\frac{1}{2} ~\times~(b_1~+~b_2)\times~h\) [Write the formula]
\(~=\frac{1}{2} ~\times~(8~+~4)\times~5\) [Substitute the values]
\(~=\frac{1}{2} ~\times~12times~5\) [Add]
\(~=30\) sq. mm [Multiply]
Therefore, the area of the trapezoid is 30 square millimeters.
Example 2: Find the height of a trapezoid whose base lengths are 12 centimeters and 8 centimeters and area is 100 square centimeters.
Solution:
As per question:
\(b_1\) = 12 cm
\(b_2\) = 8 cm
A = 100 sq. cm
\(A=\frac{1}{2} ~\times~(b_1~+~b_2)\times~h\) [Write the formula]
\(100=\frac{1}{2} ~\times~(12~+~8)\times~h\) [Substitute the values]
\(100=\frac{1}{2} ~\times~20\times~h\) [Add]
\(200=20 ~\times~h\) [Cross product]
\(10=h\) [Divide both sides by 20]
or
\(h=10\) cm
Therefore, the height of the trapezoid is 10 centimeters.
Example 3: Ethan wants to paint the top of a trapezoid shaped table. How many square centimeters will he need to paint if the parallel sides of the table measure 20 centimeters and 25 centimeters respectively, and the perpendicular distance between the sides is 22 centimeters?
Solution:
As per question:
Base \(b_1\) of the table = 20 cm
Base \(b_2\) of the table = 25 cm
Height, h of the table = 22 cm
Find the area of the table top that needs to be painted.
Since the table top is trapezoid shaped, use the formula for the area of a trapezoid:
\(A=\frac{1}{2} ~\times~(b_1~+~b_2)\times~h\) [Write the formula]
\(~=\frac{1}{2} ~\times~(20~+~25)\times~22\) [Substitute the values]
\(~=\frac{1}{2} ~\times~45\times~22\) [Add]
\(~=495\) sq. cm [Multiply]
Therefore, the area to be painted is 495 square centimeters.
The perimeter of a trapezoid is calculated by adding the measurements of its all four sides.
A trapezoid is a polygon with four sides. Hence, it is a type of quadrilateral.
A trapezoid is a type of quadrilateral, so the sum of its interior angles is 360 degrees.
Parallelograms are quadrilaterals with two sets of equal and parallel sides, but trapezoids are quadrilaterals with only one pair of parallel sides. Hence, trapezoids and parallelograms are not the same.