#### How do you find the area of planar shapes?

Area is a measure of the amount of surface a shape covers. It is usually measured in square units. So a rectangular region measuring 4 metres by 3 metres will cover 4 × 3 = 12 square metres, often written, 12 m².

It follows that the area of a rectangular region is given by Area = Base × Height.

It can be shown that the area of a parallelogram, A, is given by A = b × h, where b is the base length and h is the perpendicular height. This can be demonstrated by the following diagram, showing how a perpendicular cut can be made to form a rectanglular region.

Once this result has been establish we can demonstrate the result that the area of a triangle, A, is given by A = ½ ( b × h ), where b is the base length and h is the perpendicular height. By combining two identical triangles it is possible to form a parallelogram.

As the area of two triangles is given by b × h it follows that the area of the triangle is ½ ( b × h).

In a similar way we show that the area of a trapezium, A, is given by A = ½ ( b + t ) × h, where b is the base length, t is the top length and h is the perpendicular height.

The area of two trapeziums are ( b + t ) × h, therefore the area of a trapezium is ½ ( b + t ) × h.

It can be seen that parallelograms and triangles are both special cases of trapezium. In fact a trapezium is sometimes referred to as a truncated triangle.

By using a combination of rectangular and triangular regions and given sufficient information it is possible to find the area of any polygon.

E.g.

The area of the rectangle (square in this case) is 16 square units. The area of the 'extra' triangular regions are 2, 2 and 4½ respectively, hence the area of the triangle is 8½ square units.