Worksheet: Working with Radians

In this worksheet, we will practice using radians to find lengths, perimeters, and areas in contexts involving circles.

Q1:

A circle with radius 27 cm has a sector cut from it. The perimeter of the sector is 102 cm. What is the area of the sector?

Q2:

The area of a circular sector is 5 8 of the area of the circle. The radius is 27 cm. Find the central angle in radians giving the answer to two decimal places and the perimeter of the sector giving the answer to the nearest centimetre.

  • A 𝜃 = 3 . 9 3 r a d r a d , perimeter = 1 3 3 c m
  • B 𝜃 = 3 . 9 3 r a d r a d , perimeter = 2 3 9 c m
  • C 𝜃 = 1 . 9 6 r a d r a d , perimeter = 1 0 7 c m
  • D 𝜃 = 3 . 9 3 r a d r a d , perimeter = 1 6 0 c m
  • E 𝜃 = 1 . 9 6 r a d r a d , perimeter = 1 3 3 c m

Q3:

A sector has an arc with a measure of 7 𝜋 8 radians and a radius of 7.

Work out the length of the arc. Give your answer in terms of 𝜋 and in its simplest form.

  • A 8 𝜋
  • B 𝜋 8
  • C 8 4 9 𝜋
  • D 4 9 𝜋 8
  • E 3 4 3 𝜋 8

Work out the area of the sector. Give your answer in terms of 𝜋 and in its simplest form.

  • A 3 4 3 𝜋 1 6
  • B 3 4 3 1 6
  • C 3 4 3 𝜋 8
  • D 4 9 𝜋 8
  • E 1 6 𝜋 3 4 3

Q4:

The radius of a circle is 4 cm and the perimeter of a sector is 11 cm. Find the arc length of the sector giving the answer to the nearest centimetre.

Q5:

The area of a circular sector is 16.5 cm2 and the area of the circle that contains the sector is 9 𝜋 cm2. Find the perimeter of the circular sector.

Q6:

The arc length of a circular sector is 26 cm and the central angle is 2.36 rad. Find the area giving the answer to the nearest square centimetre and the perimeter of the sector giving the answer to the nearest centimetre.

  • A area = 1 4 3 c m , perimeter = 3 7 c m
  • B area = 3 8 0 c m , perimeter = 4 8 c m
  • C area = 7 6 0 c m , perimeter = 7 0 c m
  • D area = 1 4 3 c m , perimeter = 4 8 c m

Q7:

The area of a circular sector is 252 cm2 and the radius is 42 cm. Find the perimeter of a sector giving the answer to the nearest centimetre.

Q8:

The perimeter of a circular sector is 67 cm and the central angle is 0.31 rad. Find the radius of the sector giving the answer to the nearest centimetre.

Q9:

The area of a circular sector is 1,888 cm2 and its central angle is 1.7 rad. Find the arc length of the sector giving the answer to the nearest centimetre.

Q10:

For a circle with radius 𝑟 , one radian is defined as the measure of an arc of length 𝑟 as shown in the given figure. Knowing that the circumference of a circle is 2 𝜋 𝑟 , what is the measure of a complete revolution of a circle in radians?

  • A 𝜋 2
  • B 𝜋
  • C 2 𝜋 𝑟
  • D 2 𝜋
  • E 𝜋 𝑟

Q11:

𝑀 is a circle with radius 35 cm and 𝐴 𝐵 is an arc on 𝑀 with length 49 cm. Find the angle of the sector 𝐴 𝑀 𝐵 in radians giving the answer to one decimal place.

Q12:

Determine the angle of the circular sector that represents 7 1 8 of the circle.

Q13:

The area of a circular sector is 1 8 of the area of the circle. The radius is 20 cm. Find the central angle in radians giving the answer to two decimal places and the perimeter of the sector giving the answer to the nearest centimetre.

  • A 𝜃 = 0 . 7 9 r a d r a d , perimeter = 3 6 c m
  • B 𝜃 = 0 . 7 9 r a d r a d , perimeter = 5 1 c m
  • C 𝜃 = 0 . 3 9 r a d r a d , perimeter = 4 8 c m
  • D 𝜃 = 0 . 7 9 r a d r a d , perimeter = 5 6 c m
  • E 𝜃 = 0 . 3 9 r a d r a d , perimeter = 3 6 c m

Q14:

The area of a circular sector is 1,026 cm2 and its central angle is 1.8 rad. Find the arc length of the sector giving the answer to the nearest centimetre.

Q15:

The radius of a circle is 42 cm and the perimeter of a sector is 122 cm. Find the arc length of the sector giving the answer to the nearest centimetre.

Q16:

The perimeter of a circular sector is 139 cm and the central angle is 0.78 rad. Find the radius of the sector giving the answer to the nearest centimetre.

Q17:

The arc length of a circular sector is 32 cm and the central angle is 4 rad. Find the area giving the answer to the nearest square centimetre and the perimeter of the sector giving the answer to the nearest centimetre.

  • A area = 1 2 8 c m , perimeter = 4 0 c m
  • B area = 2 0 1 c m , perimeter = 4 8 c m
  • C area = 4 0 2 c m , perimeter = 6 4 c m
  • D area = 1 2 8 c m , perimeter = 4 8 c m

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