Area of Trapezoid Formula

Area of a trapezoid is a concept in geometry that helps to calculate the space enclosed by the unique quadrilateral. Basically it is measured in square units. In this article we will discuss in detail how to find the area of a trapezoid.

Before going to the formula of the area of the trapezoid let's understand the trapezoid.

What is a Trapezoid?

A trapezoid, which is also known as a trapezium, is a closed quadrilateral that contains a pair of parallel sides, whereas the other pair of sides is not parallel. The sides may or may not vary in length.

Properties of Trapezium

• It is a Four-sided Closed Figure with a sum of interior angles 360°.
• One pair of Parallel sides should be opposite to each other.
• One pair of non-parallel sides.
• Sum of the angles of adjacent sides is 180°.
• Diagonals of a trapezium bisect each other at an intersection.

Base of a Trapezium: The pair of parallel sides which are opposite to each other are called the base. You can call as b₁ and b₂ respectively.

Height of a Trapezium: The perpendicular distance between the two parallel lines is called as Height of the trapezium.

Area of Trapezoid Formula

If the base and height of a trapezium are given, then the area of a Trapezium can be calculated with the help of the formula: 

Area of Trapezium = 1/2 × (sum of bases) × (Height of trapezium)

Area of Trapezoid Formula Derivation

Area of a trapezium is equal to the sum of the areas of the two triangles and the area of the rectangle. Following is the derivation for calculating the area of the trapezium:

 Since we know that:

Area of Trapezium = Area of Triangle 1 + Area of Rectangle + Area of Triangle 2

Let us suppose the base of triangle 1 be B₁ and base of triangle 2 is B₂ and the height be h for both the triangles. And for rectangle assume its breadth and height be b and h.

That means,

A = (B₁ × h / 2) + b.h + (B₂ × h / 2)

Simplifying the equation, and rearranging the terms, and factoring result to:

A = h / 2[b + (B₁ + b + B₂)] ….(i)

If we assume the longer base of the trapezoid be b₁, then

b₁ = (B₁ + b + B₂)…..(ii)

Substituting (ii) in equation (i),

A = h/2(b + b₁)

Therefore, the area of a trapezeium is,

Area of Trapezium = 1 / 2 × (sum of bases) × (Height of trapezium)

Sample Problems Area of Trapezium Formula

Problem 1: Calculate the area of the trapezium in which the value of bases are 10 and 5 respectively and the height of trapezium is 2 m.

Solution

Since we know that, 

Area of trapezium = 1/2 × (sum of bases) × (Height of trapezium)

= 1/2 × (10 + 5) × 2

= 15 m²

Problem 2: Given the area of trapezium as 120 m² and height of trapezium is 6m and one of base is 4 m. Calculate the length of the other base.

Solution:

Area of trapezoid= 1/2 × (sum of bases) × (Height of trapezium)

Let value of other base is b2

Putting all the given values in the above formula, we got

120 = 1/2 × (4 + b2) × 6

120 = (4+b2) × 3

40 = (4+b2)

b2 = 36 meter

Problem 3: Calculate the area of a trapezium where the bases is 8 m and 12 m, and a height of 5 m.

Solution:

Since we know that,

Area of trapezium = 1/2× (sum of bases) ×Height of trapezium

=1/2×(8+12) ×5

= 50 m²

Problem 4: The area of a trapezium is 75 m², and the height is 5 m. What is the sum of the lengths of the parallel sides?

Solution:

Area of trapezoid= 1/2 × (sum of bases) × (Height of trapezium)

Putting all the given values in the above formula, we got

75= 1/2 × (sum of bases) × 5

75×2 = (sum of bases) × 5

sum of bases = 150/5

=30 meter

Problem 5: A trapezium has one base of 6 m, and the area is 84 m². If the height is equal to the length of one diagonal, find the length of the other base, given the height is 7 m.

Solution:

Area of trapezoid= 1/2 × (sum of bases) × (Height of trapezium)

Let value of base is b1 and b2.

here given b1=6m.

Putting all the given values in the above formula, we got

84= 1/2 × (6+ b2) × 7

168= (6+b2) × 7

24 = (4+b2)

b2 =24-6=18 meter

Problem 6: The parallel sides of a trapezium are 120 cm and 80 cm. If the height is 1.5 m, find the area of the trapezium in square meters.

Solution:

Area of trapezoid= 1/2 × (sum of bases) × (Height of trapezium)

Convert bases to meters: 120 cm = 1.2 m, 80 cm = 0.8 m

Putting all the given values in the above formula, we got

area= 1/2 × (1.2+0.8) × 1.5

area= 1/2×2 × 1.5

area = 1.5 m²

Problem 7: The ratio of the lengths of the parallel sides of a trapezium is 3:5. If the shorter base is 9 m and the height is 4 m, find the area of the trapezium.

Solution:

Area of trapezoid= 1/2 × (sum of bases) × (Height of trapezium)

Let the lengths of the bases be 3x and 5x.

Given shorter base 3x = 9 m, so x = 3 m.

Therefore, the longer base = 5x = 15 m.

Putting all the given values in the above formula, we got

area= 1/2 × (9+ 15) × 4

= 12 × 4

area= 48 m²

Problem 8: A trapezoid has parallel sides of 14 m and 10 m, and a height of 5 m. If the dimensions are scaled down by a factor of 1/2, find the area of the new trapezium.

Solution:

Let New bases = 7 m and 5 m

New height = 2.5 m

Putting all the given values in the above formula, we got

area of new trapezoid= 1/2 × (7+ 5) × 2.5

= 6 × 2.5

= 15 m²

Problem 9: A trapezium has a perimeter of 40 m, with the parallel sides measuring 12 m and 8 m. If the non-parallel sides are equal in length and the height is 6 m, find the area of the trapezium.

Solution:

Area of trapezoid= 1/2 × (sum of bases) × (Height of trapezium)

Putting all the given values in the above formula, we got

2 × non-parallel sides =40-(12+8)

= 10 m each

area of the trapezium=1/2 × (12+8) × 6

= 60 m²

Problem 10: Given the area of trapezium as 220 m² and one base is 6m and another base is 4m. Calculate the height of the trapezium.

Solution: 

To find height of trapezium, we can use the formula for the area of a trapezium:

Area = 1/2 × (b1+b2) × h

where,

• b1 is one base of Trapezium (6m)
• b2 is the other base Trapezium (4m)
• h is the height of Trapezium (which we want to find)

Given:

Area of Trapezoid = 220 m²

Substituting given values into the formula, we have:

220 = 1/2 × (6+4) × h

220 = 1/2 × 10h

220 = 5h

To find (h), we divide both sides by 5:

h = 220/5

h = 44m

So, the height of the trapezium is 44 meters.

Practice Problems on Area of Trapezoid Formula

• Find the area of a trapezium with parallel sides of length 18 cm and 24 cm, and a height of 9 cm.
• A trapezium has parallel sides of length 9 cm and 15 cm, and a height of 6 cm. What is its area?
• If the area of a trapezium is 300 cm², and the height is 15 cm, find the sum of the lengths of the parallel sides.
• Find the height of a trapezium with parallel sides of length 15 cm and 20 cm, and an area of 210 cm².
• The area of a trapezium is 180 cm², and the lengths of the parallel sides are in the ratio 2:3. If the height is 9 cm, find the lengths of the parallel sides.
• A trapezium has parallel sides of length 8 cm and 12 cm, and a height of 6 cm. What is its area?
• If the area of a trapezium is 240 cm², and the height is 12 cm, find the sum of the lengths of the parallel sides.
• Find the area of a trapezium with parallel sides of length 10 cm and 15 cm, and a height of 8 cm.
• A trapezium has an area of 360 cm², and the lengths of the parallel sides are in the ratio 3:5. If the height is 12 cm, find the lengths of the parallel sides.
• Find the area of a trapezium with parallel sides of length 20 cm and 28 cm, and a height of 11 cm.

Conclusion

By mastering the above formula and derivation we have able to find the area of trapezoid and solve various problems and also guided the importance of trapezoid concept in both academic and practical applications.