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Calculating arc length
I can calculate the arc length of a sector and the sector's perimeter.
Lesson 9 of 12
New
New
- Year 10
Calculating arc length
I can calculate the arc length of a sector and the sector's perimeter.
Lesson details
Key learning points
- Using fractions, you can calculate part of the circumference.
- This gives the length of the curved section of a sector.
- To find the perimeter, you will need to add the radius twice.
- An exact answer may be given in terms of π
Keywords
Arc - An arc is part of a curve. An arc of a circle is part of the circle’s circumference.
Sector - A sector is the region formed between two radii and their connecting arc.
Chord - A chord is any line segment joining two points on the circumference of a circle.
Radius - The radius is any line segment that joins the centre of a circle to any point on its circumference.
Circumference - The circumference of a circle is the perimeter of the circle.
Common misconception
"When finding the perimeter of a sector, I need to multiply the radius by a fraction of a full circle, just like I did with the circumference to find the arc length."
Reminder that only part (the arc length) of the formula for perimeter of a sector varies with the angle. A desmos or geogebra model can help to show this.
A circle of cheese (avoid smelly ones) or a cake with a ribbon round it can help demonstrate the "unwrappability" as you go from full circle to having a small sector removed. Focus: the addition of the two radii in the perimeter for the subtraction of a small arc length. What angle would rewrap?
Teacher tip
Licence
This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).
Lesson video
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Prior knowledge starter quiz
6 Questions
Q1.The perimeter of this shape is cm.
The perimeter of this shape is cm.
Correct Answer: 68, 68 cm, 68cm
68, 68 cm, 68cm
Q2.A circle has a diameter of 38 cm.
Which of these are correct for the circumference of this circle?
A circle has a diameter of 38 cm.
Which of these are correct for the circumference of this circle?
The circumference is $$19\pi$$ cm.
Correct answer: The circumference is $$38\pi$$ cm.
The circumference is $$38\pi$$ cm.
The circumference is $$76\pi$$ cm.
Correct answer: The circumference is 119 cm (nearest integer).
The circumference is 119 cm (nearest integer).
The circumference is 238 cm (nearest integer).
Q3.A circle has a radius of 600 cm.
Which of these are correct for the area of this circle?
A circle has a radius of 600 cm.
Which of these are correct for the area of this circle?
The area is 600$$\pi$$ cm².
The area is 600²$$\pi$$ cm.
Correct answer: The area is 600²$$\pi$$ cm².
The area is 600²$$\pi$$ cm².
The area is 360 000$$\pi$$ cm.
Correct answer: The area is 360 000$$\pi$$ cm².
The area is 360 000$$\pi$$ cm².
Q4.The perimeter of shape B is cm bigger than the perimeter of shape A.
The perimeter of shape B is cm bigger than the perimeter of shape A.
Correct Answer: 7.12, 7.12 cm
7.12, 7.12 cm
Q5.The perimeter of this shape is metres.
The perimeter of this shape is metres.
Correct Answer: 2.4, 2.4 metres, 2.40, 2.40 metres
2.4, 2.4 metres, 2.40, 2.40 metres
Q6.Use Pythagoras’ theorem to help you complete this statement. The perimeter of the triangle is cm correct to 1 d.p.
Use Pythagoras’ theorem to help you complete this statement. The perimeter of the triangle is cm correct to 1 d.p.
Correct Answer: 37.6, 37.6 cm
37.6, 37.6 cm
Assessment exit quiz
6 Questions
Q1.The sector comes from this circle.
The arc length of this sector is cm.
The sector comes from this circle.
The arc length of this sector is cm.
Correct Answer: 80, 80 cm
80, 80 cm
Q2.The sector comes from this circle.
The perimeter of the sector is cm.
The sector comes from this circle.
The perimeter of the sector is cm.
Correct Answer: 50, 50 cm
50, 50 cm
Q3.The radius of a circle is 10 cm.
The circumference of the circle is 20$$\pi$$ cm.
A semicircle is taken from this circle.
Which of these statements are correct about this semicircle?
The radius of a circle is 10 cm.
The circumference of the circle is 20$$\pi$$ cm.
A semicircle is taken from this circle.
Which of these statements are correct about this semicircle?
The arc length of the semicircle is 5 cm.
Correct answer: The arc length of the semicircle is 10$$\pi$$ cm.
The arc length of the semicircle is 10$$\pi$$ cm.
The arc length of the semicircle is 20$$\pi$$ cm.
The perimeter of the semicircle is (10$$\pi$$ + 10) cm.
Correct answer: The perimeter of the semicircle is (10$$\pi$$ + 20) cm.
The perimeter of the semicircle is (10$$\pi$$ + 20) cm.
Q4.Which of these statements are correct about the arc length of this sector?
Which of these statements are correct about the arc length of this sector?
The arc length is $${80\over360}\times30\pi$$.
Correct answer: The arc length is $${80\over360}\times60\pi$$.
The arc length is $${80\over360}\times60\pi$$.
Correct answer: The arc length is $${40\over3}\times\pi$$.
The arc length is $${40\over3}\times\pi$$.
The arc length is 41.8 cm (1 d.p.)
Correct answer: The arc length is 42 cm (nearest integer).
The arc length is 42 cm (nearest integer).
Q5.The perimeter of this sector is cm (to the nearest centimetre).
The perimeter of this sector is cm (to the nearest centimetre).
Correct Answer: 889, 889 cm
889, 889 cm
Q6.The diagram shows a sector. The chord between the endpoints of its radii is 350 cm.
The arc length of the sector is cm (to the nearest centimetre).
The diagram shows a sector. The chord between the endpoints of its radii is 350 cm.
The arc length of the sector is cm (to the nearest centimetre).
Correct Answer: 389, 389 cm
389, 389 cm
