GCSE · Maths · AQA · Spec 8300 · Foundation

Arc lengths, angles and sector areas

Cut a pizza into 8 equal slices. Each gets 1/8 of the crust, 1/8 of the cheese and a 45° point. 45 out of 360? Also 1/8. That's the trick.

Circles · Sectors

How big a slice?

angle at the centre 90°. radius 6 cm. Classification: sector. Relationship: Your slice is θ/360 of the circle: θ/360 of the rim, and θ/360 of the inside.

angle at the centre90°radius6 cmdrag B, count the slices

sector

Your slice is θ/360 of the circle: θ/360 of the rim, and θ/360 of the inside.

Drag B round the circle. The faint spokes cut the circle into 12 equal slices of 30°, and the rim dots are 10° apart. Park B at 90°: your slice holds 3 of the 12 slices and 9 of the 36 rim gaps. That's a quarter of the inside and a quarter of the rim, because 90 out of 360 is a quarter. Now try 60°, 120° and 180°.

Watch out: The angle readout here gives the smaller angle between OA and OB, so it never goes past 180°. Real sectors can be bigger than a half-turn (a pizza with one slice missing is a sector of 315°), and the same rule works for them: use the angle of the piece you actually want.

Arc length, then perimeter

Problem

A sector has radius 9 cm and angle 80°. Find (a) its arc length, in terms of π, and (b) its perimeter, to 1 decimal place.

Your turn · Sector area

Same method, different whole

A sector has radius 12 cm and angle 150°. Find its area, in terms of π.

Angle 150°, radius 12 cm. Sector area = (fraction of the circle) × (area of the whole circle).
missing step

Which line is step 2?

Fraction = 150/180 = 5/6Fraction = 150/360 = 5/12Fraction = 150/100 = 3/2

Think it through

What does the radius do?

A sector has angle 60° and radius 5 cm. Keep the angle at 60°, but make the radius 10 cm, twice as long.

What happens to the area of the sector? Pick the one closest to what you think right now.

It doubles. Twice the radius, twice the area.It stays the same. The angle hasn't changed, so neither has the slice.It becomes four times as big.

How sure are you?

I'm sureI think soI'm guessing

It doubles. Twice the radius, twice the area.

This is a common misconception

It feels right, and it IS true of the arc: the arc uses 2πr, so double r and the arc doubles. Area is different, because it uses r². Check it: radius 5 gives a whole circle of π × 5² = 25π, and radius 10 gives π × 10² = 100π. That's four times as big, not twice. The sector is 60/360 = 1/6 of each circle, so it goes from 25π/6 ≈ 13.1 cm² to 100π/6 ≈ 52.4 cm². Also four times.

It stays the same. The angle hasn't changed, so neither has the slice.

This is a common misconception

The angle fixes the FRACTION, not the size. It's still 1/6 of the circle, but it's 1/6 of a much bigger circle. One sixth of a large pizza is more pizza than one sixth of a small one. The circle's area goes from 25π to 100π, so the slice grows too, from about 13.1 cm² to about 52.4 cm².

It becomes four times as big.

This matches the evidence

Yes. The sector is still 60/360 = 1/6 of the circle, but the whole circle is now π × 10² = 100π instead of π × 5² = 25π: four times bigger, because 2² = 4. So the sector is four times bigger too, about 13.1 cm² → 52.4 cm². Doubling the ANGLE (same radius) would only double the area: that's the relationship that moves in step.

Angle and area move in step: double the angle, double the area (for angles up to 360°). Radius doesn't work like that. Area depends on r², so double the radius and the area is four times as big. Never scale the radius as if it were the angle.

Maths · Algebra

Working backwards to the angle

Same formula, different unknown. Pick a question, then step through: watch what happens to both sides on each line.

GoalThe arc of a sector is 7π cm long and the radius is 9 cm. Find the angle θ.

θ/360 × 2 × π × 9 = 7π

Put what you know into arc = θ/360 × 2πr. θ is the only unknown left.

Step 1 of 5

Put what you know into arc = θ/360 × 2πr. θ is the only unknown left.

Watch out: Divide by the WHOLE circle quantity (2πr for an arc, πr² for an area), not by r or π alone. Then check your angle by putting it back into the formula: it should give the arc or area you started with.

Spot the slip

Where does this answer go wrong?

A sector is cut from a circle of diameter 16 cm. The angle of the sector is 45°. Find the area of the sector, in terms of π.

A student's answer — which line goes wrong?

Fraction of the circle = 45/360 = 1/8
Area of the whole circle = π × 16² = 256π
Sector area = 1/8 × 256π
= 32π cm²

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A sector is just a slice of a circle, and its angle tells you exactly how big a slice.

What you need to know

A sector is a slice of a circle cut off by two radii. Its curved edge is called an arc.
The angle at the centre, θ, tells you what fraction of the whole circle the sector is: θ/360.
Arc length = (θ/360) × 2πr: that fraction of the circumference.
Sector area = (θ/360) × πr²: that fraction of the circle's area.
Perimeter of a sector = arc length + r + r.
To find a missing angle, put what you know into the arc or area formula and solve for θ.
'In terms of π' means leave π in the answer as a symbol, like 12π cm².

The big picture

A sector is a slice of a circle between two radii, and its curved edge is an arc. The angle at the centre, θ, tells you what fraction of the circle the slice is: θ/360. Take that fraction of the circumference (2πr) for the arc length, or of the area (πr²) for the sector area. The perimeter adds the two radii to the arc. To find a missing angle, solve the same formula for θ. Answers can be left exact, in terms of π.

Key points

1It's one idea, not two formulae: find the fraction θ/360, then take that fraction of the right whole (the circumference for an arc, the circle's area for a sector).

2Circumference = 2πr = πd and area of a circle = πr². Both are written with the radius, so if you're given the diameter, halve it first.

3Angle and area move in step: double the angle and the area doubles, for any angle up to 360°. Radius doesn't: double the radius and the area is four times as big.

4Change the angle and only the arc part of the perimeter changes. The two straight edges are radii, and they stay the same.

5An exact answer in terms of π is completely accurate. Round to a decimal only at the very end, and only if the question asks.

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Worked example

Problem

A Pac-Man shape is a circle of radius 4 cm with a 60° slice cut out for the mouth. Find the area of the shape, in terms of π, and its perimeter, to 1 decimal place.

⚠ Watch out

Taking the fraction of the wrong whole. Arc length is a fraction of the circumference (2πr), and sector area is a fraction of the circle's area (πr²). Mix them up, or divide by 180 instead of 360, and you get a confident-looking answer that's wrong.

🧠

Memory hook

Angle over 360 is your slice of the pie. Slice of the rim for an arc, slice of the inside for an area, and add two radii when they want the perimeter.

✓

Check yourself

A sector has radius 6 cm and angle 30°. Find its arc length and area in terms of π. (Answer: it is 1/12 of the circle: arc = π cm and area = 3π cm².)

Flashcards

(11)

What is a sector, and what is its arc?

A sector is a slice of a circle between two radii, like a pizza slice. The arc is its curved edge: a piece of the circumference.

A sector has angle θ at the centre. What fraction of the circle is it?

θ/360, because a full turn is 360°.

Formula for the arc length of a sector

Arc length = (θ/360) × 2πr: the fraction θ/360 of the circumference.

Formula for the area of a sector

Sector area = (θ/360) × πr²: the fraction θ/360 of the circle's area.

Circumference and area of a whole circle

Circumference = 2πr = πd. Area = πr².

How do you find the perimeter of a sector?

Arc length + r + r. The two straight edges are both radii.

What does 'give your answer in terms of π' mean?

Leave π as a symbol, e.g. 4π cm, instead of turning it into a decimal. It's the exact answer.

How do you find a missing sector angle from its arc length?

Write θ/360 × 2πr = arc, divide both sides by 2πr, then multiply by 360.

Same radius, angle doubled. What happens to the sector's area?

It doubles. Area moves in step with the angle (for angles up to 360°).

Same angle, radius doubled. What happens to the sector's area?

It becomes four times as big, because area depends on r² and 2² = 4. You can't scale the radius like the angle.

The question gives the diameter. What do you do before using a circle formula?

Halve it to get the radius. πr² and 2πr are both written with r.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Same method, different whole

What does the radius do?

Working backwards to the angle

Where does this answer go wrong?

More AQA GCSE Maths topics