Sectors of circles Revision Notes for AQA GCSE Maths

Sectors of circles

What is a sector?

A sector is a portion of a circle that looks like a slice of pie. It is formed by two radii (plural of radius) that extend from the centre of the circle to the edge, creating an angle at the centre.

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Think of a sector like a slice of pizza - it has two straight edges (the radii) that meet at the centre, and one curved edge (the arc) along the circumference of the circle.

Every circle can be divided into two sectors by any pair of radii:

Major sector - the larger portion of the circle
Minor sector - the smaller portion of the circle

Key formulas for sectors

When working with sectors, you need to understand that a sector represents a fraction of the whole circle. This fraction is determined by the angle at the centre.

For a sector with angle x degrees in a circle with radius r:

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Essential Sector Formulas

Area of a sector: $\text{Area} = \frac{x}{360°} \times \pi r^2$

Arc length: $\text{Arc length} = \frac{x}{360°} \times 2\pi r$

Perimeter of a sector: $\text{Perimeter} = \text{Arc length} + 2r$

Understanding the formulas

The area formula works because:

The area of a full circle is $\pi r^2$
A sector with angle $x°$ represents $\frac{x}{360°}$ of the whole circle
So we multiply the full circle area by this fraction

The arc length formula works because:

The circumference of a full circle is $2\pi r$
A sector's arc represents $\frac{x}{360°}$ of the whole circumference
So we multiply the full circumference by this fraction

The perimeter includes:

The curved edge (arc length)
Both straight edges (the two radii)

Working with sector calculations

The key to successful sector calculations is following a systematic approach and using the correct formulas for each situation.

Finding area and arc length

When calculating sector measurements, follow these essential steps:

Identify the angle ( $x$ ) and radius ( $r$ )
Substitute into the appropriate formula
Calculate using the order of operations
Round only at the final answer for accuracy

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Worked Example: Finding Arc Length and Perimeter

For a sector with angle 150° and radius 13 cm:

Step 1: Identify the given values

Angle: $x = 150°$
Radius: $r = 13$ cm

Step 2: Calculate arc length Arc length = $\frac{150°}{360°} \times 2 \times \pi \times 13$ Arc length = $\frac{5}{12} \times 26\pi = \frac{130\pi}{12} = \frac{65\pi}{6}$ cm

Step 3: Calculate perimeter Perimeter = Arc length + $2r$ Perimeter = $\frac{65\pi}{6} + 26 = 34.03... + 26 = 60$ cm (to 2 s.f.)

Finding missing angles

You can rearrange the sector formulas to find unknown angles when you know the area or arc length.

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Rearranged Formulas for Finding Angles

From area: $x = \frac{\text{Area} \times 360°}{\pi r^2}$

From arc length: $x = \frac{\text{Arc length} \times 360°}{2\pi r}$

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Worked Example: Finding Missing Angle

For a sector with area 65 cm² and radius 10 cm:

Step 1: Use the rearranged area formula $x = \frac{\text{Area} \times 360°}{\pi r^2}$

Step 2: Substitute the known values $x = \frac{65 \times 360°}{\pi \times 10^2} = \frac{65 \times 360°}{100\pi}$

Step 3: Calculate the result $x = \frac{23400°}{100\pi} = 74.5°$ (to 3 s.f.)

Exam tips

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Essential Exam Strategies

Keep answers in terms of π unless specifically asked to give a decimal approximation
Don't round intermediate steps - only round your final answer to maintain accuracy
Check your units - make sure area is in cm², length is in cm, angles in degrees
Show your working clearly - substitute values into formulas step by step
Use the correct formula - double-check whether you need area, arc length, or perimeter
Verify your answer makes sense - check if the sector represents a reasonable fraction of the circle

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Key Points to Remember:

A sector is a "slice" of a circle formed by two radii
Both area and arc length formulas start with $\frac{x}{360°}$ - the fraction of the whole circle
Perimeter of a sector = arc length + both radii
You can work backwards from area or arc length to find missing angles
Keep calculations accurate by only rounding at the final step

Sectors of circles (AQA GCSE Maths): Revision Notes