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# Area of Trapezoid – Explanation & Examples

To recall, a **trapezoid, also referred to as a trapezium**, **is a quadrilateral with one pair of parallel sides and another pair of non-parallel sides. Like square and rectangle, a trapezoid is also flat. Therefore, it is 2D.**

In a trapezoid, the parallel sides are known as the bases, while the pair of non-parallel sides are known as the legs. The perpendicular distance between the two parallel sides of a trapezium is known as a trapezoid height.

In simple words, **the base and height of a trapezoid are perpendicular to each other.**

The trapezoids can be both **right trapezoids** (two 90-degree angles) and **isosceles trapezoids** (two sides of the same length). But having one right angle is not possible because it has a pair of parallel sides, which bounds it to make two right angles simultaneously.

**In this article, you will learn:**

- How to find the area of a trapezoid,
- How to derive the trapezoid area formula and,
- How to find the area of a trapezoid using the trapezoid area formula.

## How to Find the Area of a Trapezoid?

The area of the trapezium is the region covered by a trapezium in a two-dimensional plane. It is the space enclosed in 2D geometry.

From the illustration above, a trapezoid is composed of two triangles and one rectangle. Therefore, we can calculate the area of a trapezoid by taking the sum of the areas of two triangles and one rectangle.

### Derive the trapezoid area formula

Area of a trapezium *ADEF* = (½ x *AB x FB*) + (*BC *x* FB*) + (½ x *CD x EC*)

= (¹/₂ × *AB* × *h*) + (*BC* × *h*) + (¹/₂ × *CD* × *h*)

= ¹/₂ × *h* × (*AB* + 2*BC* + *CD*)

= ¹/₂ × h × (*FE + AD*)

But, FE = b_{1} and AB = b_{2}

Hence, Area of a trapezium *ADEF*,

**= ¹/****₂**** × h × (b _{1} + b_{2}) ………………. (This is the trapezoid area formula)**

## Trapezoid Area Formula

**According to the trapezoid area formula, the area of a trapezoid is equal to half the product of the height and the sum of the two bases.**

Area = ½ x (Sum of parallel sides) x (perpendicular distance between the parallel sides).

**Area = ½ h (b _{1} + b_{2})**

Where h is the height and b_{1,} and b_{2} are the parallel sides of the trapezoid.

### How do you find the area of an irregular trapezoid?

An **irregular trapezoid** has nonparallel sides of unequal length. **To find its area, you need to find the sum of the bases and multiply it by half of the height. **

The height is sometimes missing in the question, which you can find using the Pythagorean Theorem.

### How to find the perimeter of a trapezoid?

You know perimeter is a sum of all lengths of the outer edge of a shape. Therefore, the perimeter of a trapezoid is a sum of lengths of all 4 sides.

*Example 1*

Calculate a trapezoid area whose height is 5 cm, and the bases are 14 cm and 10 cm.

****__Solution__

Let b_{1} = 14 cm and b_{2} = 10 cm

Area of trapezoid = ½ h (b_{1} + b_{2}) cm^{2}

= ½ x 5 (14 + 10) cm^{2}

= ½ x 5 x 24 cm^{2}

= 60 cm^{2}

*Example 2*

Find a trapezoid area with a height of 30 mm, and the bases are 60 mm and 40 mm.

__Solution__

Area of trapezoid = ½ h (b_{1} + b_{2}) sq. units

= ½ x 30 x (60 + 40) mm^{2}

= ½ x 30 x 100 mm^{2}

= 1500 mm^{2}

*Example 3*

The area of a trapezoid is 322 square inches. If the lengths of the two parallel sides of the trapezoid are 19 inches and 27 inches, find the trapezoid’s height.

__Solution__

Area of trapezoid = ½ h (b_{1} + b_{2}) Sq. units.

⇒ 322 square inches = ½ x h x (19 + 27) Sq. inches

⇒ 322 square inches = ½ x h x 46 Sq. inches

⇒ 322 = 23h

Divide both sides by 23.

h = 14

So, the height of the trapezoid is 14 inches.

*Example 4*

Given that the height of a trapezoid is 16 m and one base’s length is 25 m. Calculate the dimension of the other base of the trapezoid if its area is 352 m^{2}.

__Solution__

Let b_{1} = 25 m

Area of trapezoid = ½ h (b_{1} + b_{2}) sq. units

⇒ 352 m^{2} = ½ x 16 m x (25 m + b_{2}) sq. units

⇒ 352 = 8 x (25 + b_{2})

⇒ 352 = 200 + 8b_{2}

Subtract 200 on both sides.

⇒ 152 = 8b_{2}

Divide both sides by 8 to get;

b_{2} = 19

Therefore, the length of the other base of the trapezoid is 19 m.

*Example 5*

Calculate the area of the trapezoid shown below.

__Solution__

Since the legs (non-parallel sides) of the trapezoid are equal, then the height of the trapezoid can be calculated as follows;

To get the two triangles’ base, subtract 15 cm from 27 cm and divide by 2.

⇒ (27 – 15)/2 cm

⇒ 12/2 cm = 6 cm

12^{2} = h^{2 }+ 6^{2}By Pythagorean theorem, the height (h) is calculated as;

144 = h^{2} + 36.

Subtract 36 on both sides.

h^{2} = 108.

h = 10.39 cm.

Hence, the height of the trapezoid is 10.39 cm.

Now, calculate the area of the trapezoid.

Area of trapezoid = ½ h (b_{1} + b_{2}) Sq. units.

= ½ x 10.39 x (27 + 15) cm^{2}.

= ½ x 10.39 x 42 cm^{2}.

= 218.19 cm^{2}.

*Example 6*

One base of a trapezium is 10 m more than the height. If the other base is 18 m and the trapezoid area is 480 m^{2}, find the height and base of the trapezoid.

__Solution__

Let the height = x

Other base is 10 m than the height = x + 10.

Area of trapezoid = ½ h (b_{1} + b_{2}) Sq. units.

By substitution,

480 = ½ * x * (x + 10 + 18)

480 = ½ *x * (x + 28)

Use the distributive property to remove the parentheses.

480 = ½x^{2} + 14x

Multiply each term by 2.

960 = x^{2} + 28x

x^{2} + 28x – 960 = 0

Solve the quadratic equation to get;

x = – 48 or x = 20

Substitute the positive value of x in the equation of height and base.

Height: x = 20 m.

The other base = x + 10 = 10 + 20 = 30 m.

Therefore, the other base and height of the trapezoid are 30 and 20 m, respectively.

*Practice Problems*

*Practice Problems*

- Find the area of a trapezoid, which has the parallel bases of lengths 9 units and 12 units, and the height is 15 units.
- For a trapezoidal figure, the sum of the parallel bases is 25 m, and the height is 10 m. Determine the area of this figure.
- Consider a trapezoid of area
*112b*square ft, where*b*is the shorter base length. What is the height of this trapezoid if the lengths of two parallel bases are such that one base is two times the other base?