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

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.

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

Finding a length in composite shapes

Maths

In this unit, pupils have derived the formula for the circumference and area of a circle. In the future unit, this knowledge is combined with knowledge of fractions to find the arc length and area of a sector.

 In the prior unit, pupils learnt how to derive and use the formula for the area of a triangle and trapezium. In this unit, this knowledge is called upon to find the volume of different prisms and area of composite shapes.

Perimeter, area and volume

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?

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.

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?

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 114 cm². Which of these statements about the shape are correct?

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

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

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

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

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?

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

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. $$r=$$ {{}}

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

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

Find the positive value of $$x$$ that is a solution to the equation: $$3{\times}7{\times}2x^2 = 3{\times}56$$. $$x=$${{}}

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

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.

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.

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