New
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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.
Lesson 9 of 16
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.
Lesson details
Key learning points
- The diameter can be found from the area of circles or parts of circles by rearranging the formula.
- The radius can be found from the area of circles or parts of circles.
- 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 halve 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.
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 intuitively, with the one exception of the three-quarter-circle.
Teacher tip
Equipment
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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Prior knowledge starter quiz
6 Questions
Q1.Find the positive value of $$x$$ that is a solution to the equation: $$x^2 + 4 = 85$$. $$x=$$ .
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
9, x = 9, x=9
Q2.Select the fully factorised form of this expression: $$42r + 56$$.
Select the fully factorised form of this expression: $$42r + 56$$.
$$2(21r+28)$$
$$7(6r+8)$$
$$14(7r)$$
Correct answer: $$14(3r+4)$$
$$14(3r+4)$$
$$9(3r+4)$$
Q3.Find the positive value of $$x$$ that is a solution to the equation: $$3{\times}7{\times}2x^2 = 3{\times}56$$. $$x=$$
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
2, x = 2, x=2
Q4.Select the fully factorised form of this expression: $$2kx^2 + 3kx$$.
Select the fully factorised form of this expression: $$2kx^2 + 3kx$$.
$$k(2x^2 + 3x)$$
$$x(2kx + 3k)$$
Correct answer: $$kx(2x + 3)$$
$$kx(2x + 3)$$
$$kx(2x + 3k)$$
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.
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
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.
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
17, 17 cm, 17cm
Assessment exit quiz
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?
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
128$$\pi$$ cm²
402.12$$\pi$$ cm²
16384$$\pi$$ cm²
Correct answer: 402.12 cm²
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?
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 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 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?
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?
Area of the square is $$r$$
Correct answer: Area of the square is $$r^2$$
Area of the square is $$r^2$$
Correct answer: Area of the quarter-circle is $${1\over4}{\times}{\pi}{\times}r^2$$
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 $${1\over4}{\times}{\pi}{\times}r^2 + r^2$$
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?
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?
$$r\left( {{1\over4}{\times}{\pi}+1}\right) = 216$$
$$r\left( {{1\over4}{\times}{\pi}{\times}r+r}\right) = 216$$
Correct answer: $$r^2\left( {{1\over4}{\times}{\pi}+1}\right) = 216$$
$$r^2\left( {{1\over4}{\times}{\pi}+1}\right) = 216$$
$$r{\times}\left({1\over4}{\times}4.1416...\right) = 216$$
Correct answer: $$r^2{\times}(1.785...) = 216$$
$$r^2{\times}(1.785...) = 216$$
Q5.The area of this composite shape is: ($$9216\pi + 16384$$) cm². Find the value of $$r$$. $$r=$$ .
The area of this composite shape is: ($$9216\pi + 16384$$) cm². Find the value of $$r$$. $$r=$$ .
Correct Answer: 128, 128 cm, 128cm
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. $$r=$$
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. $$r=$$
Correct Answer: 23, 23 cm, 23cm
23, 23 cm, 23cm
