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 $=s^2$=`s`2
Rectangle		Area $=lw$=`lw`
Parallelogram		Area $=bh$=`bh`
Trapezium		Area $=\frac{1}{2}\left(a+b\right)h$=12(`a`+`b`)`h`
Circle		Area $=\pi r^2$=π`r`2
Sector		Area =$\frac{\theta}{360}\pi r^2$`θ`360π`r`2

Practice using the formula for these building block shapes before moving on to composite shapes.

Practice questions

Question 1

Find the area of the solid triangle with base length $8$8 m and perpendicular height $10$10 m shown below.

An obtuse triangle whose base sits horizontally. The obtuse angle is at the bottom left vertex. A dashed line extends the base to a vertical dashed line at a right angle (indicated by a small square). The top of the vertical dashed line meets the top corner of the main triangle. The vertical dashed line is labeled $10$ m. The base side of the main triangle is labeled $8$ m.

Question 2

Find the area of this trapezium using the rule

$\text{Area }=\frac{1}{2}\times\left(\text{side a}+\text{side b}\right)\times\text{height h }$Area =12×(side a+side b)×height h

A trapezium with two parallel bases has measurements of $10$ $mm$ on its top base and $6$ $mm$ on the bottom base. A gray vertical dashed line measuring $2$ $mm$ is drawn from the top base to the bottom base. A gray small square is drawn in the intersection indicating that the angle at their intersection is a right angle. The two legs on the left and right sides connecting the bases are not labeled.

Question 3

A circle has a radius of $12$12 $mm$ .

What is the exact area of the circle?
What is the area of the circle rounded to two decimal places?

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 4

Find the area of the given shape.

A two-dimensional inverted L-shaped figure is displayed. The top horizontal line is labeled $13$ $m$ . On the right portion, the upper vertical side is labeled $4$ $m$ , while the bottom vertical side is labeled $12$ $m$ . The bottom horizontal line is labeled $5$ $m$ .

Question 5

Find the total area of the figure shown.

A composite figure composed of a triangle and a parallelogram. The triangle has a horizontal base that measures $16$16 $m$ as labeled. A vertical dashed line from the apex of the triangle to its base measures $4$4 $m$ . The parallelogram has vertical left and right sides that both measure $13$13 $m$ as labeled. A horizontal dash line that measures $8$8 $m$ connect the vertical sides, indicating the length of the base of the parallelogram. The parallelogram has slanted top and bottom sides. The parallelogram sits on top of the triangle such that the slanted bottom side of the parallelogram is the same as the right side of the triangle.

Question 6

Find the area of the shaded figure shown, correct to two decimal places.

Outcomes

ACMGM018

solve practical problems requiring the calculation of perimeters and areas of circles, sectors of circles, triangles, rectangles, parallelograms and composites

5.04 Area of composite shapes

Areas of simple shapes

Practice questions

Question 1

Question 2

Question 3

Areas of composite shapes

Worked example

Practice questions

Question 4

Question 5

Question 6

Outcomes

ACMGM018

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