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VCE 11 General 2023

10.05 Area of composite shapes

Lesson

A composite shape is a shape that can be broken up into smaller more recognisable shapes. For example, this shape is a square and a triangle combined.

Finding areas of composite shapes requires us to be able to break up the shape into recognisable components.

 

Cutting up shapes

It may be easier to see small shapes that make up a large one ("adding" shapes together):

Sometimes it's easier to see a large shape with a bit missing ("subtracting" shapes). It's good to get practice at both. So, the same shape could be a large rectangle with a small rectangle cut out of it.

What shapes can you see in these pictures? Practice breaking the composite shape up into smaller parts, or looking for a larger shape with a piece cut out of it. Remember to look for shapes already studied such as rectangles, squares, triangles or parallelograms.

 

Finding areas of composite shapes

To find the areas of composite shapes, being able to identify the shapes is only the first step. The next is calculating the areas of the parts. Consider the following two examples from the shapes above.

Worked examples

example 1

Find the area of the composite shape below.

Think: What shapes can we see? We can break up the composite shape into two triangles.

Do: The first triangle is given below.

The base measurement is $3-1=2$31=2 cm and the height is $3.2$3.2 cm. So the area of the triangle is:

Area of triangle $=$= $\frac{1}{2}\times b\times h$12×b×h
  $=$= $\frac{1}{2}\times2\times3.2$12×2×3.2
  $=$= $3.2$3.2 cm2

 

The other triangle is given below.

The base measurement is $3$3 cm and the height measurement is $3$3 cm. So the area of the triangle is:

Area of triangle $=$= $\frac{1}{2}\times b\times h$12×b×h
  $=$= $\frac{1}{2}\times3\times3$12×3×3
  $=$= $4.5$4.5 cm2

 

So the total area of the composite shape is $3.2+4.5=7.7$3.2+4.5=7.7 cm2.

example 2

A backyard garden needs to have turf laid. The shape and dimensions of the garden are indicated in the picture below. Find the area of the turf required.


Think: What shapes can we see? We can think of the turf as a rectangle with a triangular piece taken out as shown below.

Do:

The outside rectangle has area:

Area $=$= $l\times w$l×w
  $=$= $12\times9$12×9
  $=$= $108$108 m2

The corner triangle has area:

Area $=$= $\frac{1}{2}\times b\times h$12×b×h
  $=$= $\frac{1}{2}\times\left(12-5\right)\times\left(9-4\right)$12×(125)×(94)
  $=$= $\frac{1}{2}\times7\times5$12×7×5
  $=$= $17.5$17.5 m2

So the total area is the area of the rectangle minus the area of the triangle:

Total area $=$= $108-17.5$10817.5
  $=$= $90.5$90.5 m2

 

Practice questions

Question 1

Find the total area of the figure shown.

A composite figure made up of two triangles.

Question 2

Annulus

An annulus is a doughnut shape where the area is formed by two circles with the same centre.

Area of an annulus

$\text{Area of an annulus}=\text{area of larger circle}-\text{area of smaller circle}$Area of an annulus=area of larger circlearea of smaller circle

Practice question

Question 3

Find the area of the shaded region in the following figure, correct to one decimal place.

 

Sectors

Area of sectors

$\text{Area of a circle}=\pi r^2$Area of a circle=πr2

$\text{Area of a semi-circle}=\frac{1}{2}\pi r^2$Area of a semi-circle=12πr2

$\text{Area of a quarter-circle}=\frac{1}{4}\pi r^2$Area of a quarter-circle=14πr2

Always consider the fraction of the circle. For example consider a quarter circle–the angle of this sector is $90^\circ$90°. The fraction of the circle is $\frac{90}{360}=\frac{1}{4}$90360=14. Therefore to calculate the area of a quarter of a circle, we calculate the whole circle and divide by $4$4.

Practice questions

Question 4

Calculate the area of the following figure, correct to one decimal place.

A circle with a 90-degree sector is removed, as indicated by a blue box. The radius of the circle measures 2$cm$cm

Question 5

Find the area of the shaded region in the following figure, correct to one decimal place.

A figure of a square with a quarter-circle cut out of it such that the bottom-right corner of the square intersects with the corner of the quarter-circle. The bottom-right corner has a small square indicating a right angle. The top and left sides of the square each has a tick mark. The right side of the square is divided into two segments with $4$4-cm segment on top of the $20$20-cm segment. The $20$20-cm segment is also the radius of the quarter-circle. 

Question 6

Find the area of the shaded region in the following figure, correct to one decimal place.

Outcomes

U2.AoS4.5

the perimeter and areas of triangles (using several methods based on information available), quadrilaterals, circles and composite shapes, including arcs

U2.AoS4.11

calculate the perimeter and areas of triangles (calculating the areas of triangles in practical situations using the rules A=1/2 bh, A=1/2 ab sin(c) or A=\sqrt{s(s-a)(s-b)(s-c)} where s=(a+b+c)/2

U2.AoS4.12

use quadrilaterals, circles and composite shapes including arcs and sectors in practical situations

U2.AoS4.13

calculate the perimeter, areas, volumes and surface areas of solids (spheres, cylinders, pyramids and prisms and composite objects) in practical situations, including simple uses of Pythagoras’ in three dimensions

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