Trapezoid – Definition, Types, Properties and Formulas

Trapezoid is another name of trapezium and is a quadrilateral in which one pair of opposite sides are parallel. It is a quadrilateral and follows all the properties of a quadrilateral.

In this article we have covered, Trapezoid definition, types of trapezoid, properties of trapezoid, formulas of trapezoid and related example in detail.

What is a Trapezoid?

It is a quadrilateral that has one pair of parallel sides and another set of non-parallel sides. The parallel sides are known as bases. The other two non-parallel sides are called lateral sides. Altitude or height is the shortest distance between the two parallel sides.

A trapezoid is shown in the image below:

Types of Trapezoid

There are three types of trapezoids that are:

1. Isosceles Trapezoid: Lengths of non-parallel sides are equal in Isosceles Trapezoid.
2. Right-Angled Trapezoid: It has a set of right angles.
3. Scalene Trapezoid: No sides are equal in Scalene Trapezoid.

Properties of a Trapezoid

Various properties of a Trapezoid are:

• Trapezoid has only one pair of parallel sides.
• Sum of all angles of a Trapezoid is 360^o.
• For a Isosceles Trapezoid, both base angles are congruent , non parallel sides are congruent and diagonals are congruent.
• Median is parallel to both the bases and is the average length of the bases in an trapezoid.
• Adjacent angles of a trapezoid sum to 180^o.

Trapezoid Formulas

Formulas for trapezoid consists of Area of Trapezoid and Perimeter of Trapezoid, let’s learn the same in detail.

Area of Trapezoid

Let us consider a trapezoid  PQRS, SR and PQ are parallel sides and ST is the shortest distance between them which is perpendicular and is denoted with h.

Let‘s divide the trapezoid into segments. That gives us two triangles PTS and UQR and a rectangle TURS. Now if we find the areas of all three and sum them, we get the area of the Trapezoid.

Area of triangle, PTS = 1/2 (base × height) 

Area of triangle, PTS = 1/2 (PT × ST)

= 1/2 (xh)……(1)

On writing the area of rectangle TURS = length × breadth  

Area of rectangle TURS = bh……(2)

In writing the area of triangle UQR = 1/2 (base × height) 

Area of triangle UQR = 1/2 ( yh)…..(3)

In summing up the three areas i.e. equations (1), (2), and (3), we get the area of the trapezium.

On adding all the three equations, 

Area of Trapezoid = 1/2 (xh) + bh + 1/2 ( yh)

on taking 1/2 (h) common, we can write the equation as 

Area of Trapezoid = (1/2) h (x + 2b + y)

Area of Trapezoid = (1/2) h ( x + b + b + y)

From the figure, we can observe that summing x, b, and y gives a. So,

Area of Trapezoid = (1/2) × h × (a+b)

or

Area of Trapezoid = (1/2) × (height) × (sum of the parallel sides)

Perimeter of Trapezoid

For finding the perimeter, the lengths of all the sides are to be added. 

Perimeter of Trapezoid = Sum of lengths of parallel and non-parallel sides

For any trapezoid ABCD with sides, AB = a, BC = b, CD = c, DA = a, its perimeter is:

Perimeter of Trapezoid = a + b + c + d

Hence, the Perimeter of a trapezoid is the sum of lengths of parallel sides and the sum of lengths of non-parallel sides. 

Read More,

• Rhombus
• Triangle
• Quadrilaterals

Example on Trapezoid Formulas

Example 1: Find the area of trapezoid if the bases are 10 cm and 16 cm. The shortest distance between the parallel sides is 8 cm.

Solution:

Given lengths of bases,

• a = 10 cm
•  b = 16cm
• height h = 8cm

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

= (1/2) 8 (10+16)

= 140 cm²

Example 2: If the area of a trapezoid is given as 240 cm² and the sum of lengths of parallel sides is given as 30 cm, find the height of the trapezoid.

Solution:

Given,

• Area = 240 cm²
• (a+b) = 30 cm

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

240  = (1/2) h (30)

h = (240 × 2) / 30

h = 16 cm

Hence, the height of trapezoid is 16 cm

Example 3: Find perimeter of trapezoid if lengths of the sides are 15cm, 6cm, 12 cm, and 8 cm respectively.

Solution: 

Given,

• a = 15 cm
• b = 6 cm
• c = 12 cm
• d = 8 cm

Perimeter of Trapezoid = Sum of lengths of parallel sides and sum of lengths of non parallel sides

P = a + b + c + d

P = 15+6+12+8

P = 41 cm

Example 4: If the perimeter of a trapezium is 48 cm and the sum of parallel sides is 26 cm and the length of the Third side is 10 cm. Find the length of the fourth side.

Solution: 

Given,

• a+b = 26 cm
• c =10 cm
• Perimeter = 48 cm

Perimeter of Trapezoid = Sum of lengths of parallel sides and the sum of lengths of nonparallel sides

Perimeter of Trapezoid = a + b + c + d

48 = 26 + 10 + d 

d = 48- 26 – 10

= 12 cm

Hence the length of the fourth side is 12 cm

Example 5: If the length of a parallel side is greater than the other by 6 cm and the area of a trapezoid is 240 cm². Find the lengths of the parallel sides If the shortest distance between the parallel sides is 14 cm.

Solution:

Given,

• Area = 240 cm²
• Height = 14 cm

Let a parallel side a = x

According to given condition

b = x + 6

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

240 = (1/2) (14) (x + x + 6)

(240×2) / 14 = 2x + 6 

2x = 34.28 – 6

2x = 28.28

x = 14.14

Hence a = 14.14 cm

b = x + 6

b = 14.14 + 6

b = 20.14 cm

Hence 14.14 cm and 20.14 cm are the lengths of the parallel sides

FAQs on Trapezoid

Is trapezoid a trapezium?

Yes a trapezoid and trapezium are same quadrilateral.

What is a trapezoid in maths?

Trapezoid in maths is a quadrilateral in which one pair of opposite sides are parallel.

What are the 3 types of trapezoid?

The 3 types of trapezoid are:

• Right Trapezoid
• Isosceles Trapezoid
• Scalene Trapezoid

Do all trapezoids have 4 equal sides?

Yes, as all trapezoid are quadrilateral all trapezoids have four sides.

What is the formula to a trapezoid?

Area of a trapezoid is found using the formula, A = ½ (a + b) h