Sectors of circles Revision Notes for Edexcel GCSE Maths

Sectors of circles

What is a sector?

A sector is a portion of a circle that looks like a slice of pie. When you draw two straight lines (called radii) from the centre of a circle to its edge, you create two sectors.

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Understanding sectors is essential for many circle calculations. Think of sectors as "slices" - just like cutting a pizza into pieces, each slice represents a fraction of the whole circle.

There are two types of sectors:

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

Key formulas for sectors

When working with a sector that has an angle x° and comes from a circle with radius r, you need to remember that the sector represents a fraction of the whole circle.

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Key concept: The fraction is $\frac{x}{360°}$

This fraction is the foundation of all sector calculations - it tells you what portion of the complete circle your sector represents.

Area of a sector

The area formula combines the sector fraction with the total circle area:

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

This works because you're finding what fraction of the whole circle area $(πr²)$ your sector represents.

Arc length

The arc is the curved edge of the sector.

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

This calculates what fraction of the whole circumference $(2πr)$ your arc represents.

Perimeter of a sector

The perimeter includes the curved arc plus the two straight edges (radii).

$\text{Perimeter of sector} = \text{Arc length} + \text{radius} + \text{radius}$

Worked example: Finding perimeter

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Worked Example: Finding Sector Perimeter

A minor sector has an angle of 150° and radius 13 cm. Find the perimeter.

Step 1: Calculate the arc length

Arc length = $\frac{150°}{360°} \times 2\pi \times 13$
Arc length = $\frac{150°}{360°} \times 26\pi$
Arc length = 34.0392... cm

Step 2: Add the two radii

Perimeter = Arc length + radius + radius
Perimeter = 34.0392... + 13 + 13
Perimeter = 60 cm (to 2 significant figures)

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Exam tip: Don't round until your final answer for better accuracy. Keep full calculator values throughout your working.

Finding missing angles

You can rearrange the sector formulas to find missing angles when given the area or arc length. This reverse-engineering approach is common in exam questions.

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Worked Example: Finding Missing Angle

A sector has area 65 cm² and radius 10 cm. Find the angle.

Step 1: Use the area formula

Area of sector = $\frac{x}{360°} \times \pi r^2$
$65 = \frac{x}{360°} \times \pi \times 10^2$
$65 = \frac{x}{360°} \times 100\pi$

Step 2: Solve for x

$x = \frac{65 \times 360°}{100\pi}$
$x = 74.4845...°$
$x = :success[74.5° (to 3 significant figures)]$

Exam tips

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Essential Exam Strategies:

Always check whether you need to find a major or minor sector
Remember that angles in sector formulas must be in degrees, not radians
Keep your calculator answers until the end, then round to the required accuracy
For perimeter questions, don't forget to add both radii to the arc length
Double-check your formula selection - area vs arc length vs perimeter

Key takeaways

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Key Points to Remember:

A sector is a fraction of a circle, like a slice of pie
The fraction is $\frac{\text{angle}}{360°}$
Area of sector = $\frac{x}{360°} \times \pi r^2$
Arc length = $\frac{x}{360°} \times 2\pi r$
Perimeter = arc length + 2 × radius
You can rearrange these formulas to find missing angles or measurements
Always work with degrees unless specifically told otherwise

Sectors of circles (Edexcel GCSE Maths): Revision Notes