What is a Trapezoid?
A trapezoid is a quadrilateral with exactly one pair of opposite sides that are parallel. The pair of opposite sides that are parallel are usually the top and the bottom sides of a trapezoid.
Properties of a Trapezoid
- The parallel sides are called bases
- The non-parallel sides are called legs
- The perpendicular distance between the bases is called height or altitude
- The two consecutive angles with a common side as the base are called base angles
- The angles formed on the same leg, between the parallel sides, are supplementary
- The sum of all interior angles is 360°
Area of a Trapezoid
The area A of trapezoid is one half of the product of its height h and sum of its bases b\(_1\) and b\(_2\).
A = \(\frac{1}{2}\) x h x (b\(_1\) + b\(_2\) )
It must be noted that the area of geometric shapes is measured in square units.
Types of Trapezoids
There are three types of trapezoids:
- Right trapezoid
- Isosceles trapezoid
- Scalene trapezoid
1. Right trapezoid : A trapezoid with one pair of adjacent right angles is called a right trapezoid.
2. Isosceles trapezoid : A trapezoid whose non parallel sides, called legs, are congruent or equal in length is called an isosceles trapezoid. The base angles of an isosceles trapezoid are equal.
3. Scalene trapezoid : A trapezoid that has no sides that are equal in length is a scalene trapezoid.
Solved Trapezoid Examples
Example 1: Find the area of the given trapezoid.
Solution:
A = \(\frac{1}{2}\) x h x (b\(_1\) + b\(_2\)) [Write the formula]
A = \(\frac{1}{2}\) x 4 x (5 + 6) [Substitute]
A = 22
So, the area of the given trapezoid is 22 square inches.
Example 2: A garden is in the shape of a trapezoid as shown in the image. The area of the garden is 10000 square meters. Find the shortest distance between the parallel sides of the garden?
Solution: The shortest distance between the parallel sides is the height of the trapezoid.
We can use the formula for the area of a trapezoid to find the height.
A = \(\frac{1}{2}\) x h x (b\(_1\) + b\(_2\)) [Write the formula]
10000 = \(\frac{1}{2}\) x h x (150 + 100) [Substitute 10000 for A, 150 for b\(_1\) and 100 for b\(_2\)]
10000 x 2 = h x 250 [Multiply by 2 on each side.]
\(\frac{20000}{250}\) = h [Divide each side by 250]
80 = h [Solve]
So, the shortest distance between parallel sides of the trapezoid shaped garden is 80 meters.
Example 3: Find the value of x.
Solution :
The given trapezoid shows that AD = BC, so this is an isosceles trapezoid.
∠ADC = ∠BCD [Base angles of an isosceles trapezoid are equal]
65 = x + 15 [Substitute 65 for ∠ADC and x + 15 for ∠BCD.]
65 – 15 = x + 15 – 15 [Subtract 15 on both sides]
50 = x [Solve]
So, the value of x is 50.
A parallelogram has both pairs of opposite sides parallel but a trapezoid has exactly one pair of opposite sides that are parallel. So, trapezoids are not parallelograms.
A trapezoid with legs that are equal in length is called an isosceles trapezoid.
When the legs of a trapezoid are parallel to each other, which indicates that both pairs of opposite sides are parallel, the trapezoid then becomes a parallelogram.
A quadrilateral with exactly one pair of opposite sides parallel is called a trapezoid.
The area of a trapezoid is one half of the product of its height and the sum of the length of its bases.