Worksheet: Central Angles in a Circle

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In this worksheet, we will practice identifying a central angle in a circle and finding its measure using its properties.

Q1:

In the given figure, the circle has center 𝑀and radius 7 cm. Given that 𝐴𝐵=14.9cm, find 𝑀𝐵, rounded to the nearest tenth, and 𝑚∠𝐴𝑀𝐶.

  • A 𝑀 𝐵 = 1 4 . 9 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 5 1 ∘
  • B 𝑀 𝐵 = 1 6 . 5 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 1 9 ∘
  • C 𝑀 𝐵 = 1 3 . 2 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 1 9 ∘
  • D 𝑀 𝐵 = 1 6 . 5 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 5 1 ∘

Q2:

How many 1∘ angles are in a circle?

Q3:

How many 90∘ turns are needed to make a complete turn?

Q4:

How many 45∘ angles does it take to make a full turn?

Q5:

Jacob turns one-third of a full turn. How many degrees is this?

Q6:

This circle is divided into 3 equal parts. What is the angle of each part?

Q7:

What is the value of the central angle of a circular sector when a circular segment and a circular sector have the same area?

Q8:

Find 𝑚∠𝐴𝑀𝐵.

Q9:

𝐴 , 𝐵 , and 𝐷 are three points on a circle with radius 54 cm and center 𝑀. The point 𝐶 is the intersection of 𝑀𝐷 and 𝐴𝐵 and we have 𝑀𝐶⟂𝐴𝐵 and 𝑀𝐶=42cm. Find the central angle of the major circular segment 𝐴𝐹𝐵 giving the answer to the nearest second.

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

Q10:

A circle has a circumference of 16𝜋 units. Find, in degrees, the measure of the central angle of an arc with a length of 3𝜋 units.

Q11:

In the figure, given that the circumference is 96 and 𝐴𝐵=12, find 𝜃.

Q12:

What is the measure of half a circle?

Q13:

Circle 𝑀 has radius 32 and 𝐴𝐵=56.5. Determine 𝑚∠𝐷𝑀𝑋 and 𝐷𝐸 to the nearest hundredth.

  • A 1 0 8 ∘ , 56.50
  • B 1 4 4 ∘ , 28.25
  • C 1 2 6 ∘ , 16.97
  • D 5 4 ∘ , 15.03

Q14:

In the given figure, the circle has center 𝑀and radius 9.9 cm. Given that 𝐴𝐵=17.1cm, find 𝑀𝐵, rounded to the nearest tenth, and 𝑚∠𝐴𝑀𝐶.

  • A 𝑀 𝐵 = 1 7 . 1 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 0 4 ∘
  • B 𝑀 𝐵 = 1 9 . 8 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 6 6 ∘
  • C 𝑀 𝐵 = 1 3 . 9 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 6 6 ∘
  • D 𝑀 𝐵 = 1 9 . 8 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 0 4 ∘

Q15:

In the given figure, the circle has center 𝑀and radius 8.5 cm. Given that 𝐴𝐵=16.5cm, find 𝑀𝐵, rounded to the nearest tenth, and 𝑚∠𝐴𝑀𝐶.

  • A 𝑀 𝐵 = 1 6 . 5 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 3 3 ∘
  • B 𝑀 𝐵 = 1 8 . 6 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 3 7 ∘
  • C 𝑀 𝐵 = 1 4 . 1 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 3 7 ∘
  • D 𝑀 𝐵 = 1 8 . 6 c m , 𝑚 ∠ 𝐴 𝑀 𝐶 = 1 3 3 ∘

Q16:

The given figure shows a circle with center 𝑂.

Which of the following is a central angle?

  • A ∠ 𝐵 𝐴 𝐶
  • B ∠ 𝐶 𝐷 𝐵
  • C ∠ 𝑂 𝐶 𝐷
  • D ∠ 𝐵 𝑂 𝐶
  • E ∠ 𝐵 𝐷 𝐶

Which of the following is an inscribed angle?

  • A ∠ 𝐵 𝐴 𝐶
  • B ∠ 𝑂 𝐶 𝐷
  • C ∠ 𝐵 𝐷 𝐶
  • D ∠ 𝐶 𝐷 𝐵
  • E ∠ 𝐵 𝑂 𝐶

Which of the following is a circumscribed angle?

  • A ∠ 𝐵 𝑂 𝐶
  • B ∠ 𝑂 𝐶 𝐷
  • C ∠ 𝐵 𝐴 𝐶
  • D ∠ 𝐵 𝑂 𝐶
  • E ∠ 𝐵 𝐷 𝐶

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