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Year 10
Higher

Calculating arc length

I can calculate the arc length of a sector and the sector's perimeter.

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New
New
Year 10
Higher

Calculating arc length

I can calculate the arc length of a sector and the sector's perimeter.

Download all resources
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Share activities with pupils
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Lesson details

Key learning points

  1. Using fractions, you can calculate part of the circumference.
  2. This gives the length of the curved section of a sector.
  3. To find the perimeter, you will need to add the radius twice.
  4. 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 (2024), 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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Starter quiz

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6 Questions

Q1.
The perimeter of this shape is cm.
An image in a quiz
Correct Answer: 68, 68 cm, 68cm
Q2.
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 $$76\pi$$ cm.
Correct answer: 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?
The area is 600$$\pi$$ cm².
The area is 600²$$\pi$$ cm.
Correct answer: The area is 600²$$\pi$$ cm².
The area is 360 000$$\pi$$ cm.
Correct answer: The area is 360 000$$\pi$$ cm².
Q4.
The perimeter of shape B is cm bigger than the perimeter of shape A.
An image in a quiz
Correct Answer: 7.12, 7.12 cm
Q5.
The perimeter of this shape is metres.
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Correct Answer: 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.
An image in a quiz
Correct Answer: 37.6, 37.6 cm

Exit quiz

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6 Questions

Q1.
The sector comes from this circle. The arc length of this sector is cm.
An image in a quiz
Correct Answer: 80, 80 cm
Q2.
The sector comes from this circle. The perimeter of the sector is cm.
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Correct Answer: 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 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 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.
Q4.
Which of these statements are correct about the arc length of this sector?
An image in a quiz
The arc length is $${80\over360}\times30\pi$$.
Correct answer: The arc length is $${80\over360}\times60\pi$$.
Correct answer: 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).
Q5.
The perimeter of this sector is cm (to the nearest centimetre).
An image in a quiz
Correct Answer: 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).
An image in a quiz
Correct Answer: 389, 389 cm