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Area of a Trapezium (or Trapezoid)

A trapezium (or trapezoid) has one pair of opposite sides that are parallel.

Let us look at a trapezium with the two parallel sides to be a and b, and the height of the trapezium to be h. This trapezium can be split into a rectangle and a a triangle as shown below –

area of trapezium_1

Area of a trapezium  =  area of the rectangle  +  area of the triangle

=  ah  +  1/2(b – a)h

=  ah  +  1/2bh  –  1/2ah

=  1/2ah  +  1/2bh

=  1/2h(a + b)

So the formula to calculate the area of a trapezium is  A  =  1/2h(a + b)

 

If we are given a trapezium (or trapezoid) with height h = 4 cm, and the lengths of the two parallel sides, namely a and b = 8 cm and 14 cm respectively, we can obtain the area of a trapezium using the formula, or by calculating the areas of the rectangle and the triangle separately and adding them up.

area of trapezium_2

Area of trapezium  =  area of rectangle  +  area of triangle

=  8 x 4  +  1/2x 6 x 4

=  32 + 12

=  44 cm2

Using the formula, we can also obtain the area of a trapezium to be;

A  =  1/2h(a + b)

=  1/2x 4 x (8 + 14)

=  44 cm2

 

area of trapezium_5Example 1 : Find the area of the trapezium given in this diagram. Use both the formula and the split diagram approaches to get the area.

Applying the formula A  =  1/2h(a + b)

Area  =  1/2x 5 x (6 + 16)

=  55 cm2

Splitting the trapezium we get –

area of trapezium_3

Area of the trapezium  =  area of rectangle  +  area of triangle

=  6 x 5  +  1/2x 5 x 10

=  30 + 25

=  55 cm2

Generally, we tend to use the formula to obtain the area of a trapezium – it is easy and quick.

 

area of trapezium_4Example 2 : What is the area of this trapezium?

Area  =  1/2h(a + b)

=  1/2x 6 x (14 + 3)

=  51 cm2

So the area of the trapezium is 51 cm2.