Question

A regular hexagon is inscribed in a circle as shown. Determine the measure of arc AB.
A) 30°
B) 45°
C) 60°
D) 80°

Answer 1

A regular hexagon is inscribed in a circle as shown. The measure of arc AB is given below.

Construction: Join AP.

Interior angles of any polygon is given by the formula,

(n - 2) × 180° / n

A regular hexagon has 6 sides. Therefore, n = 6

= (6 - 2) × 180° / 6

= 4 × 180° / 6

= 120°

Therefore, ∠ A = ∠ B = 120°

The line BP and AP divides the angles ∠ A and  ∠ B into two equal angles i.e., ∠ Ap = ∠ Bp = 60°

Now consider,

In Δ APB,

∠ Ap + ∠ Bp + ∠ P = 180°

60° + 60° + ∠ P = 180°

120° + ∠ P = 180°

∠ P = 180° - 120°

∴ ∠ P = 60°

As we have, ∠ APB = 60°

Now consider the inscribed angle theorem,

The measure of inscribed angle is half the measure of its intercepted arc.

⇒ ∠ = 1/2 (m arc)

∠ APB = 1/2 (m arc AB)

60° = 1/2 (m arc AB)

120° = (m arc AB)

Therefore the measure of arc AB  is 120°

Answer 2

Answer:

C. 60

Step-by-step explanation:

To find the central angle, we can divide 360 by 6 because a hexagon has six sides. Once you have solved that, you will get 60 and since 60 is the measure of the central angle, each arc will also equal 60 degrees.