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

=  ah  +  (b – a)h

=  ah  +  bh  –  ah

=  ah  +  bh

=  h(a + b)

So the formula to calculate the area of a trapezium is  A  =  h(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  =  area of rectangle  +  area of triangle

=  8 x 4  +  x 6 x 4

=  32 + 12

=  44 cm2

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

A  =  h(a + b)

=  x 4 x (8 + 14)

=  44 cm2

Example 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  =  h(a + b)

Area  =  x 5 x (6 + 16)

=  55 cm2

Splitting the trapezium we get –

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

=  6 x 5  +  x 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.

Example 2 : What is the area of this trapezium?

Area  =  h(a + b)

=  x 6 x (14 + 3)

=  51 cm2

So the area of the trapezium is 51 cm2.