Trapezoid Formula

A trapezoid is also known as a trapezium is a four-sided polygon or a quadrilateral. It has one set of opposite sides which are parallel and a set of non-parallel sides. The parallel sides are known as the bases and the non-parallel sides are known as the legs of the trapezoid.

There are different types of trapezoids: isosceles trapezoid, right trapezoid, and scalene trapezoid.

•  A trapezoid with two non-parallel sides of the same length is called an isosceles trapezoid.
• A right trapezoid is a trapezoid that has at least two right angles.
• A right isosceles trapezoid is a trapezoid that is simultaneously a right trapezoid and an isosceles trapezoid. In Euclidean geometry, such trapezoids are automatically rectangles.
• A scalene trapezoid has neither of the sides or angles of equal measure.

Let's look into the Trapezoid formulas in detail.

What is Trapezoid Formula?

We will be learning the following Trapezoid Formula

• Perimeter of a Trapezoid
• Area of a Trapezoid

Formula to Calculate Perimeter of a Trapezoid

The perimeter of a trapezoid is defined as the sum of all its sides or the complete boundary of the trapezoid. Consider a trapezoid ABCD as shown below with side measures a,b,c, and d. Let's look into the Trapezoid Formula

The perimeter of the trapezoid formula is calculated by finding the sum of all the sides i.e, AB + BC + CD + DA

Perimeter of a Trapezoid = Sum of all the sides = a + b + c + d

where, a, b,c, and d are the sides of the trapezoid. 

Formula to Calculate Area of a Trapezoid

The area is defined as the area or region occupied by the trapezoid. It is half the product of the sum of its bases and the distance between them. It is the Consider a trapezoid ABCD as shown below.

The area of a trapezoid is given by:

Area of a Trapezoid  = (1/2) × h × (a + b)

where,

• a = Shorter base
• b = Longer base
• h = Height or distance between the two bases

Let us see the applications of the trapezoid formula in the following solved examples.

Examples Using Trapezoid Formula

Example 1: If the perimeter of a trapezoid is 60 units and three of its sides are 15 units, 20 units, and 16 units respectively, find the measure of the fourth side using the Trapezoid Formula.
Solution: 

Given: Perimeter = 60 units, a = 15 units, b = 20 units, c = 16 units, d = ?

We know that, according to Trapezoid Formula,

Perimeter of a trapezoid = Sum of all the sides

⇒ a + b + c + d = 60

⇒ 15 + 20 + 16 + d = 60

⇒ d = 9 units

Answer: Thus, the fourth side measures 9 units.

Example 2: Using area of trapezoid formula, find the area of a trapezoid whose bases are 19 units and 15 units and height is 8 units.
Solution:

Given: 

a = 17 units

b = 19 units

h = 8 units

We know that, according to trapezoid formula,

Area of a trapezoid =  h(a + b) / 2

= 8 (15 + 19) / 2

= 4 × 34 

= 136 units²

Answer: Thus, the area of the trapezoid is 136 units²

Example 3: If the area of the trapezoid is 120 inches and the lengths of the bases are 12 inches and 20 inches, find the height of the trapezoid using the trapezoid formula?

Solution: 

Let us assume that the bases are a and b, and the height of the trapezoid is h. Using the given information, we have to find height which is the distance between the bases. Let us substitute all these values in the area of a trapezoid formula:

Area of trapezoid formula, A = [(a + b)/2] × h

120 = [(20 + 12)/2] × h

120 = 16 × h

h = 7.5 inches

Therefore, the height of the trapezoid is 7.5 inches.

Answer: The height of the trapezoid is 7.5 inches.