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Year 8

Finding a length in composite shapes

I can solve area problems of composite shapes involving whole and/or part circles where the area is known and the radius or diameter must be found.

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New
New
Year 8

Finding a length in composite shapes

I can solve area problems of composite shapes involving whole and/or part circles where the area is known and the radius or diameter must be found.

Download all resources
Share activities with pupils

Link copied to clipboard

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Lesson details

Key learning points

  1. The diameter can be found from the area of circles or parts of circles by rearranging the formula.
  2. The radius can be found from the area of circles or parts of circles.
  3. The diameter or radius can be found from the area of composite shapes by reasoning and rearranging the formula.

Keywords

  • Sector - A sector is the region formed between two radii and their connecting arc.

  • Diameter - The diameter of a circle is any line segment that starts and ends on the edge of the circle and passes through the centre of the circle.

  • Radius - The radius is any line segment that joins the centre of a circle to its edge.

Common misconception

If I want to find the radius of a semicircle from its area, I half the area first.

You must double the area. This calculates the area of a full circle, whose radius can be found by first dividing by π, then square rooting.


To help you plan your year 8 maths lesson on: Finding a length in composite shapes, download all teaching resources for free and adapt to suit your pupils' needs...

Finding the radius of more general sectors will be covered in subsequent units; this lesson will only cover the most common unitary sectors whose full circle can be calculated intiutively, with the one exception of the three-quarter-circle.
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Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Starter quiz

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6 Questions

Q1.
Find the positive value of $$x$$ that is a solution to the equation: $$x^2 + 4 = 85$$. $$x=$$ .
Correct Answer: 9, x = 9, x=9
Q2.
Select the fully factorised form of this expression: $$42r + 56$$.
$$2(21r+28)$$
$$7(6r+8)$$
$$14(7r)$$
Correct answer: $$14(3r+4)$$
$$9(3r+4)$$
Q3.
Find the positive value of $$x$$ that is a solution to the equation: $$3{\times}7{\times}2x^2 = 3{\times}56$$. $$x=$$
Correct Answer: 2, x = 2, x=2
Q4.
Select the fully factorised form of this expression: $$2kx^2 + 3kx$$.
$$k(2x^2 + 3x)$$
$$x(2kx + 3k)$$
Correct answer: $$kx(2x + 3)$$
$$kx(2x + 3k)$$
Q5.
A circle has a radius of 1.2 cm. Find the area of this circle. Give your answer in cm, rounded to 2 decimal places. Area = cm².
Correct Answer: 4.52, 4.52 cm², 4.52cm²
Q6.
The area of a circle is 908 cm². Use the formula: $$area = {\pi} {\times}r^2$$ to form an equation and solve this equation to find the radius of the circle, rounded to the nearest integer. cm.
Correct Answer: 17, 17 cm, 17cm

Exit quiz

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6 Questions

Q1.
A quarter-circular sector is cut out of a circle. The quarter-circle has an area of 32$$\pi$$ cm². What was the area of the original circle?
8$$\pi$$ cm²
Correct answer: 128$$\pi$$ cm²
402.12$$\pi$$ cm²
16384$$\pi$$ cm²
Correct answer: 402.12 cm²
Q2.
A semi-circular sector is cut out of a circle. The semicircle has an area of 98$$\pi$$ cm². Which of these statements are true about the original circle?
The original circle has an area of 49$$\pi$$ cm².
Correct answer: The original circle has an area of 196$$\pi$$ cm².
The original circle has a radius of 7 cm.
Correct answer: The original circle has a radius of 14 cm.
The original circle has a diameter of 14 cm.
Q3.
This composite shape is composed of a quarter-circle and a square. The area of the composite shape is 114 cm². Which of these statements about the shape are correct?
An image in a quiz
Area of the square is $$r$$
Correct answer: Area of the square is $$r^2$$
Correct answer: Area of the quarter-circle is $${1\over4}{\times}{\pi}{\times}r^2$$
Correct answer: Area of the whole shape is $${1\over4}{\times}{\pi}{\times}r^2 + r^2$$
Area of the whole shape is $${1\over4}{\times}{\pi}{\times}r+ r $$
Q4.
This composite shape is composed of a quarter-circle and a square. The area of the composite shape is 216 cm². Which of these equations are correct steps in the method to find the radius?
An image in a quiz
$$r\left( {{1\over4}{\times}{\pi}+1}\right) = 216$$
$$r\left( {{1\over4}{\times}{\pi}{\times}r+r}\right) = 216$$
Correct answer: $$r^2\left( {{1\over4}{\times}{\pi}+1}\right) = 216$$
$$r{\times}\left({1\over4}{\times}4.1416...\right) = 216$$
Correct answer: $$r^2{\times}(1.785...) = 216$$
Q5.
The area of this composite shape is: ($$9216\pi + 16384$$) cm². Find the value of $$r$$. $$r=$$ .
An image in a quiz
Correct Answer: 128, 128 cm, 128cm
Q6.
This composite shape is composed of two quarter-circles and a square. The area of this composite shape is 1360 cm². Find the value of $$r$$, rounded to the nearest integer. $$r=$$
An image in a quiz
Correct Answer: 23, 23 cm, 23cm