Finding a length in composite shapes | Maths | Quizalize & Oak National Academy

Starter quiz

Find the positive value of $x$ that is a solution to the equation: $x^2 + 4 = 85$ . $x=$ ______.

'9' ✓
Select the fully factorised form of this expression: $42r + 56$ .

$2(21r+28)$

$7(6r+8)$

$14(7r)$

$14(3r+4)$ ✓

$9(3r+4)$
Find the positive value of $x$ that is a solution to the equation: $3{\times}7{\times}2x^2 = 3{\times}56$ . $x=$ ______

'2' ✓
Select the fully factorised form of this expression: $2kx^2 + 3kx$ .

$k(2x^2 + 3x)$

$x(2kx + 3k)$

$kx(2x + 3)$ ✓

$kx(2x + 3k)$
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.

'4.52' ✓
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.

'17' ✓

Exit quiz

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

128 $\pi$ cm² ✓

402.12 $\pi$ cm²

16384 $\pi$ cm²

402.12 cm² ✓
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.

The original circle has an area of 196 $\pi$ cm². ✓

The original circle has a radius of 7 cm.

The original circle has a radius of 14 cm. ✓

The original circle has a diameter of 14 cm.
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 $r$

Area of the square is $r^2$ ✓

Area of the quarter-circle is ${1\over4}{\times}{\pi}{\times}r^2$ ✓

Area of the whole shape is ${1\over4}{\times}{\pi}{\times}r^2 + r^2$ ✓

Area of the whole shape is ${1\over4}{\times}{\pi}{\times}r+ r$
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?

$r\left( {{1\over4}{\times}{\pi}+1}\right) = 216$

$r\left( {{1\over4}{\times}{\pi}{\times}r+r}\right) = 216$

$r^2\left( {{1\over4}{\times}{\pi}+1}\right) = 216$ ✓

$r{\times}\left({1\over4}{\times}4.1416...\right) = 216$

$r^2{\times}(1.785...) = 216$ ✓
$The area of this composite shape is: (<Math>9216\pi + 16384</Math>) cm². Find the value of <Math>r</Math>. <Math>r=</Math><span class="blank">______</span>.$

The area of this composite shape is: ( $9216\pi + 16384$ ) cm². Find the value of $r$ . $r=$ ______.

'128' ✓
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. $r=$ ______

'23' ✓

Worksheet

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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.

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.

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.