Area of a Trapezium

# Area of a Trapezium

## Area of a Trapezium

A trapezium is a two-dimensional 4-sided figure whose one pair of opposite sides are parallel and the other pair is not parallel. The non-parallel sides of the trapezium are called the legs of the trapezium. The area of a trapezium is the space occupied by the boundary of the trapezium. If a trapezium covers an area of 15 sq. cm, then its area is 15 sq. cm.
The area of a trapezium is defined as the number of unit squares that can fit in the trapezium. The area of a trapezium is measured in sq. cm, sq. m, sq. in, sq. km, etc. These are the units of the area of a trapezium.
Every time, it is not possible to divide the trapezium into square units. In that case, we need a formula to calculate the area of a trapezium. Let us see the formula to calculate the area of a trapezium.

## Area of a Trapezium Formula

Area of the trapezium = ½ × (a + b) × h

## How to Derive Area of a Trapezium Formula

In the given figure, ABCD is a trapezium with AB || DC. Let us draw perpendiculars

DP and CQ on AB, from D and C, respectively.

Let us consider DP = CQ = h, i.e., the height of the trapezium ABCD.

From the given figure,

Area of the trapezium ABCD = Area of ΔDAP + Area of rectangle DPQC + Area of ΔCQB = ½ AP × h + PQ × h + ½ QB × h = ½ × h × (AP + 2PQ + QB)

= ½ × h × {(AP + PQ + QB) + PQ}

= ½ × h × (AB + PQ)                     [Since AP + PQ + QB = AB]

= ½ × h × (AB + CD)

= ½ × h × (sum of the parallel sides)

So, Area of the trapezium = ½ × altitude × sum of the parallel sides

= ½ × distance between the parallel sides × sum of the parallel sides

Let the lengths of the parallel sides be a and b.

Then, Area of the trapezium = ½ × (a + b) × h

## How To Calculate Area of Trapezium?

Follow the below steps to find the area of a trapezium:

• Step 1: Find and note down the lengths of the parallel sides (bases) of the trapezium from the given data.
• Step 2: Calculate the sum of the lengths of the parallel sides.
• Step 3: Multiply the value of the sum of parallel sides by height of the trapezium and again by 1/2.
• Step 4: Write the answer in square units. This is the required area of the trapezium.

### Area of Trapezium Examples

Example 1: The parallel sides of a trapezium are 25 cm and 15 cm long. If the distance between them is 12 cm, find the area of the trapezium.

Solution:

Here, a = 25 cm and b = 15 cm

Distance between the parallel sides (h) = 12 cm

Area of the trapezium = ½ × (a + b) × h

= ½ × (25 + 15) × 12

= ½ × 40 × 12 = 240 sq. cm

Example 2: Find the distance between the parallel sides of a trapezium whose parallel sides are 16 cm and 14 cm and the area of the trapezium is 270 sq. cm.

Solution:

Here, a = 16 cm and b = 14 cm

Given: Area of the trapezium = 270 sq. cm

Distance between the parallel sides (h) = ?

Area of the trapezium = ½ × (a + b) × h

270 = ½ × (16 + 14) × h

270 = ½ × 30 × h

270 = 15 × h

h = 270/15

h = 18 cm

Example 3: Find the area of a trapezium whose parallel sides are 18 cm and 16 cm, respectively, and the distance between them is 10 cm.

Solution:

Here, a = 18 cm and b = 16 cm

Distance between the parallel sides (h) = 10 cm

Area of the trapezium = ½ × (a + b) × h

= ½ × (18 + 16) × 10

= ½ × 34 × 10 = 170 sq. cm

Example 4: Find the height of a trapezium whose parallel sides are 24 cm and 20 cm, respectively, and the area of the trapezium is 330 sq. cm.

Solution:

Here, a = 24 cm and b = 20 cm

Given: Area of the trapezium = 330 sq. cm

Distance between the parallel sides (h) = ?

Area of the trapezium = ½ × (a + b) × h

330 = ½ × (24 + 20) × h

330 = ½ × 44 × h

330 = 22 × h

h = 330/22

h = 15 cm

To find the area of a rhombus and the area of a parallelogram, click the following links:

Area of a Rhombus

Area of a Parallelogram

Area of a Square

Area of a Rectangle

Area of a Circle