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.

  • A9πœ‹
  • B3πœ‹
  • C2πœ‹
  • D18πœ‹
  • E6π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.

  • A2π‘Ÿ
  • B2πœ‹π‘ŸοŠ¨
  • C2πœ‹π‘Ÿ
  • Dπœ‹π‘Ÿ
  • Eπœ‹π‘ŸοŠ¨

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

  • A180πœƒ
  • B360πœƒ
  • Cπœƒ360
  • Dπœƒπ‘Ÿ
  • Eπœƒ180

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

  • Aπœƒ360Γ—2πœ‹π‘Ÿ
  • Bπœƒ360Γ—π‘Ÿ
  • Cπœƒ180Γ—2πœ‹π‘Ÿ
  • Dπœƒ360Γ—πœ‹π‘Ÿ
  • Eπœƒ360Γ—2π‘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:

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.

  • Aπœƒπ‘Ÿ
  • Bπ‘Ÿπœƒ
  • Cπ‘Ÿπœƒ2
  • D2π‘Ÿπœƒ
  • Eπ‘Ÿπœƒ

Q6:

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

Q7:

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.

  • A3πœ‹8
  • B3πœ‹2
  • C3πœ‹4
  • D4πœ‹3
  • E2πœ‹3

Q8:

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.

  • A330β€²53β€²β€²βˆ˜
  • B6617β€²22β€²β€²βˆ˜
  • C9926β€²4β€²β€²βˆ˜
  • D661β€²47β€²β€²βˆ˜
  • E1499β€²6β€²β€²βˆ˜

Q9:

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.

Q10:

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

Q11:

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.

  • A11435β€²30β€²β€²βˆ˜
  • B23454β€²46β€²β€²βˆ˜
  • C24622β€²19β€²β€²βˆ˜
  • D1263β€²3β€²β€²βˆ˜

Q12:

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.

  • A5πœ‹ cm
  • B92πœ‹ cm
  • C54πœ‹ cm
  • D52πœ‹ cm

Q13:

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.

  • A537β€²30β€²β€²βˆ˜
  • B5717β€²45β€²β€²βˆ˜
  • C738β€²22β€²β€²βˆ˜
  • D616β€²56β€²β€²βˆ˜

Q14:

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.

Q15:

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.

Q16:

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

Q17:

What is the length of the arc of a circle of radius π‘Ÿ that is given by 23 of the circumference?

  • A13πœ‹π‘Ÿ
  • B43πœ‹π‘Ÿ
  • C3πœ‹π‘Ÿ
  • D32πœ‹π‘Ÿ
  • E23πœ‹π‘Ÿ

Q18:

A circle has a central angle measuring 7028β€²48β€²β€²βˆ˜ which subtends an arc of length 21.18 cm. Find the radius of the circle giving the answer to the nearest centimeter.

Q19:

A circle has a radius of 14.49 cm. Find the central angle, in radians, that subtends an arc of length 8.23 cm, giving the answer to two decimal places.

Q20:

The perimeter of a circular sector is 19 cm and the central angle is 0.375 rad. Find the arc length giving the answer to the nearest centimetre.

Q21:

A circle has a radius of 1.5 cm, where the central angle subtending an arc is 1.88 rad. Find the length of the arc, giving the answer to one decimal place.

Q22:

Calculate the length of an arc on Earth’s surface that subtends an angle of 5 minutes at Earth’s center, knowing that 1 minute is equal to 160 of a degree. Take the Earth’s radius to be 3,960 miles.

Q23:

A circle has a central angle measuring 1.515 rad which subtends an arc of length 25.36 cm. Find the radius of the circle to the nearest centimeter.

Q24:

An arc on a circle with a radius of 12 has a length of 14. Determine the arc’s measure, giving your answer in radians as a fraction in its simplest form.

  • A105πœ‹
  • Bπœ‹105
  • C76
  • D210πœ‹
  • E67

Q25:

The radius of a circle is 40 cm and the perimeter of a sector is 106 cm. Find the central angle in degrees giving the answer to the nearest second and in radians giving the answer to one decimal place.

  • A1837β€²16β€²β€²βˆ˜, 0.7 rad
  • B3714β€²32β€²β€²βˆ˜, 0.7 rad
  • C1837β€²16β€²β€²βˆ˜, 0.3 rad
  • D3714β€²32β€²β€²βˆ˜, 0.3 rad

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