Arc length and sector area

About this lesson

Geometry › Circles

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Arc length and sector area

Warm-up

Three quick questions from earlier lessons.

Discover

A 60∘60^\circ slice is cut from a round pie. What fraction of the whole pie is it?

  1. 16\tfrac{1}{6}
  2. 160\tfrac{1}{60}

Key idea

Arcs and sectors are fractions of the circle.

For a central angle θ\theta in degrees: arc length =θ360⋅2πr= \dfrac{\theta}{360} \cdot 2\pi r; sector area =θ360⋅πr2= \dfrac{\theta}{360} \cdot \pi r^2.

Worked example

r=6r = 6, θ=120∘\theta = 120^\circ.

120360=13\tfrac{120}{360} = \tfrac{1}{3}
One third.
Arc =4π= 4\pi
⅓ · 12π.
Sector =12π= 12\pi
⅓ · 36π.

Your turn, guided

Your turn: r=10r = 10, θ=90∘\theta = 90^\circ. The arc length (in terms of π).

5π5\pi
¼ · 20π.

Practice ★

  1. A circle has radius 8 cm. Find the length of an arc with central angle 152∘152^\circ (exact, in terms of π).Not to scale
  2. A circle has radius 12 cm. Find the length of an arc with central angle 289∘289^\circ (exact, in terms of π).Not to scale
  1. Find the length of the arc cut off by a 90∘90^\circ central angle in a circle of radius 12 (exact).Not to scale

Watch out!

Practice ★★

  1. Find the area of a sector with radius 5 cm and central angle 188∘188^\circ (exact, in terms of π).Not to scale
  2. Find the area of a sector with radius 10 cm and central angle 147∘147^\circ (exact, in terms of π).Not to scale
  1. Boss problem. A slice of pizza is a sector of radius 6 cm with central angle 50∘50^\circ. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

  1. A circle has radius 8 cm. Find the length of an arc with central angle 152∘152^\circ (exact, in terms of π).Not to scale
  2. A circle has radius 12 cm. Find the length of an arc with central angle 289∘289^\circ (exact, in terms of π).Not to scale
  3. A circle has radius 12 cm. Find the length of an arc with central angle 91∘91^\circ (exact, in terms of π).Not to scale

Summary card

Arcs and sectors

Arc =θ3602πr= \tfrac{\theta}{360} 2\pi r, sector =θ360πr2= \tfrac{\theta}{360}\pi r^2.

Watch out: circumference for arcs, area for sectors.

End of the lesson

Next lesson: Radian measure and arc length Back to the chapter

Answers: Arc length and sector area

A

  1. 304π45\frac{304 \pi}{45}
  2. 289π15\frac{289 \pi}{15}

B

  1. 6π6 \pi

C

  1. 235π18\frac{235 \pi}{18}
  2. 245π6\frac{245 \pi}{6}

D

  1. 5π5 \pi

E

  1. 304π45\frac{304 \pi}{45}
  2. 289π15\frac{289 \pi}{15}
  3. 91π15\frac{91 \pi}{15}

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