5.04 Area of composite shapes

Areas of simple shapes

The following table gives a summary of formulas for the areas of triangles, quadrilaterals (squares, rectangles, parallelograms and trapezia), sectors and circles. Being familiar with each will be useful when finding the area of composite shapes.

Triangle		Area $=\frac{1}{2}bh$=12​bh
Square		Area $=s^2$=s2
Rectangle		Area $=lw$=lw
Parallelogram		Area $=bh$=bh
Trapezium		Area $=\frac{a+b}{2}h$=a+b2​h
Circle		Area $=\pi r^2$=πr2
Sector		Area =$\frac{\theta}{360}\pi r^2$θ360​πr2

 

Areas of composite shapes

Composite shapes consist of multiple circles, sectors, or polygons, all patched together or cut out from each other. We can find the areas of composite shapes by adding or subtracting the areas of simple shapes as appropriate.

 

Worked example

Find the area of the following composite shape.

Think: We can start with a rectangle that covers the whole figure, then look for the simple shapes we can cut away from this rectangle to get back to the composite shape. These will be the areas we will subtract from the area of the rectangle.

The figure below shows that we will be subtracting the areas of two triangles (triangle B and triangle C), and two parallelograms (parallelogram D and parallelogram E) from the area of one large rectangle (rectangle A).

Do: Combining the formulas for the area of each simple shape, we have:

Total area	$=$=	Area of rectangle $-$− Area of triangles $-$− Area of parallelograms
	$=$=	$14\times13-\frac{1}{2}\times6\times13-\frac{1}{2}\times6\times13-2\times5-8\times3$14×13−12​×6×13−12​×6×13−2×5−8×3
	$=$=	$14\times13-6\times13-2\times5-8\times3$14×13−6×13−2×5−8×3
	$=$=	$182-78-10-24$182−78−10−24
	$=$=	$70$70

So the area of the composite shape is $70$70 m².

Reflect: Labelling each constituent shape can be useful to keep track of which shape is being added and which shape is being subtracted.

 

Practice questions

Question 1

Question 2

Question 3