Worksheet: Arc Lengths and Sectors

In this worksheet, we will practice finding the arc length, the perimeter, and the area of a circular sector using their formulas.

Q1:

Find the length of the blue arc given the radius of the circle is 8 cm. Give the answer to one decimal place.

Q2:

An arc has a measure of 2πœ‹3 radians and a radius of 9. Work out the length of the arc, giving your answer in terms of πœ‹, in its simplest form.

  • A 3 πœ‹
  • B 9 πœ‹
  • C 2 πœ‹
  • D 6 πœ‹
  • E 1 8 πœ‹

Q3:

For the given figure, 𝑂 is the center of the circle and πœƒ is the measure of arc 𝑙.

Write down an expression for the circumference of the circle.

  • A πœ‹ π‘Ÿ 
  • B 2 π‘Ÿ
  • C 2 πœ‹ π‘Ÿ
  • D πœ‹ π‘Ÿ
  • E 2 πœ‹ π‘Ÿ 

If πœƒ is measured in degrees, what fraction of the circle’s circumference is arc 𝑙?

  • A 3 6 0 πœƒ
  • B πœƒ 1 8 0
  • C 1 8 0 πœƒ
  • D πœƒ 3 6 0
  • E πœƒ π‘Ÿ

Write an expression for the length of arc 𝑙, given that πœƒ is measured in degrees.

  • A πœƒ 3 6 0 Γ— π‘Ÿ
  • B πœƒ 3 6 0 Γ— πœ‹ π‘Ÿ
  • C πœƒ 3 6 0 Γ— 2 πœ‹ π‘Ÿ
  • D πœƒ 1 8 0 Γ— 2 πœ‹ π‘Ÿ
  • E πœƒ 3 6 0 Γ— 2 π‘Ÿ

Q4:

Circle 𝑀 has a radius of 12 cm where the length of 𝐢𝐡 is 16 cm. Find the length of arc 𝐢𝐡 giving the answer to two decimal places.

Q5:

Given that 𝐴𝑀=11cm, find the length of 𝐴𝐡𝐢 rounded to the nearest integer.

Q6:

Write an expression for the length of an arc whose measure is πœƒ radians, knowing that the expression for the length of an arc in degrees is 2πœ‹π‘Ÿπœƒ360.

  • A π‘Ÿ πœƒ
  • B π‘Ÿ πœƒ 2
  • C πœƒ π‘Ÿ
  • D 2 π‘Ÿ πœƒ
  • E π‘Ÿ πœƒ

Q7:

If π‘šβˆ π΄=76∘ and the radius of the circle equals 3 cm, find the length of the major arc 𝐡𝐢.

Q8:

An arc has a measure of πœ‹8 radians and a radius of 6. Work out the length of the arc, giving your answer in terms of πœ‹, in its simplest form.

  • A 3 πœ‹ 8
  • B 3 πœ‹ 2
  • C 2 πœ‹ 3
  • D 3 πœ‹ 4
  • E 4 πœ‹ 3

Q9:

What is the length of the arc subtended by a central angle of 261∘ on a circle of radius π‘Ÿ?

  • A 2 9 2 0 πœ‹ π‘Ÿ
  • B 2 9 1 0 πœ‹ π‘Ÿ
  • C 2 9 4 0 πœ‹ π‘Ÿ
  • D 2 9 8 0 πœ‹ π‘Ÿ

Q10:

An arc on a circle with a radius of 50 has a length of 115. Determine the arc’s measure to the nearest tenth of a degree.

Q11:

A circle has a radius of 7.22 cm. Find the central angle that subtends an arc of length 12.53 cm, giving the answer to the nearest second.

  • A 6 6 1 β€² 4 7 β€² β€² ∘
  • B 3 3 0 β€² 5 3 β€² β€² ∘
  • C 1 4 9 9 β€² 6 β€² β€² ∘
  • D 6 6 1 7 β€² 2 2 β€² β€² ∘
  • E 9 9 2 6 β€² 4 β€² β€² ∘

Q12:

The length of an arc in a circle is 1.2π‘Ÿ where π‘Ÿ is the radius of the circle. Find the central angle subtending the arc in radians giving the answer to one decimal place.

Q13:

An arc in a circle measures 16πœ‹π‘Ÿ. What angle does it subtend?

Q14:

What angle is subtended by an arc of length 20 in a circle of circumference 80?

Q15:

An arc covers 29 of a circle’s circumference and the circle has a radius of 78 cm. Find the measure and the length of the arc using 227 as an approximation for πœ‹, rounding the length to the nearest hundredth..

  • A 4 0 ∘ , 54.45 cm
  • B 8 0 ∘ , 108.91 cm
  • C 8 0 ∘ , 54.45 cm
  • D 4 0 ∘ , 108.91 cm

Q16:

The length of an arc in a circle is 2π‘Ÿ where π‘Ÿ is the radius of the circle. Find the central angle subtending the arc in degrees giving the answer to the nearest second.

  • A 2 3 4 5 4 β€² 4 6 β€² β€² ∘
  • B 2 4 6 2 2 β€² 1 9 β€² β€² ∘
  • C 1 1 4 3 5 β€² 3 0 β€² β€² ∘
  • D 1 2 6 3 β€² 3 β€² β€² ∘

Q17:

The radius of a sundial is 15 cm and the shadow changes at a rate of 15∘ every hour. Find in terms of πœ‹ the arc length of the rotation of the shadow after 2 hours.

  • A 5 4 πœ‹ cm
  • B 5 2 πœ‹ cm
  • C 5 πœ‹ cm
  • D 9 2 πœ‹ cm

Q18:

The radius of a circle is 15 cm and the arc length of a sector is 16 cm. Find the central angle giving the answer to the nearest second.

  • A 7 3 8 β€² 2 2 β€² β€² ∘
  • B 6 1 6 β€² 5 6 β€² β€² ∘
  • C 5 7 1 7 β€² 4 5 β€² β€² ∘
  • D 5 3 7 β€² 3 0 β€² β€² ∘

Q19:

A circle has a central angle of 6454β€²58β€²β€²βˆ˜ which subtends an arc of length 4πœ‹ cm. Find the diameter of the circle to the nearest centimeter.

Q20:

Calculate the length of an arc on Earth’s surface that subtends an angle of 7 minutes at Earth’s center knowing that 1=160minutedegrees of a degree. Take Earth’s radius to be 3,960 miles.

Q21:

The area of a circular sector is 815.1 cm2 and the central angle is 62∘. Find the arc length of the sector giving the answer to the nearest centimeter.

Q22:

In the given figure, which of the following would represent the minor arc passing through 𝐡 and 𝐢?

  • A βƒ–     βƒ— 𝐡 𝐢
  • B 𝐡 𝐢
  • C οƒͺ 𝐢 𝐡
  • D 𝐡 𝐢
  • E οƒͺ 𝐡 𝐢

Q23:

Given that ⃖⃗𝐴𝐢 is a tangent to the circle 𝑀, where it touches it at the point 𝐴,𝐡𝑀=36cm, and 𝐴𝐢=54cm, find the length of 𝐴𝐷.

Q24:

𝑀 is a circle of radius 19 cm. Determine, to the nearest hundredth, the length of 𝐡𝐷.

Q25:

Determine, to the nearest hundredth, the length of π‘π‘Œ.

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