Question

consider a regular hexagon inscribed in circle c with radius r. regular hexagons have six congruent sides and six congruent angles. in the figure, angle cab measures 30°. what is m∠acb? what is the length of segment ab? what is the perimeter of the hexagon? the perimeter of the hexagon is how circumference of the circle. the circumference of the circle is

Explanation:

Step1: Recall properties of a regular hexagon inscribed in a circle

A regular hexagon inscribed in a circle can be divided into six equilateral triangles with side - length equal to the radius of the circle. In \(\triangle ABC\), since the hexagon is regular and inscribed in the circle with center \(C\), \(CA = CB=\text{radius}=r\) and \(AB\) is a side of the hexagon. So, \(AB = r\).

Step2: Find \(\angle ACB\)

The central angle of a regular hexagon corresponding to each side is \(\frac{360^{\circ}}{6}=60^{\circ}\), so \(\angle ACB = 60^{\circ}\).

Step3: Calculate the perimeter of the hexagon

Since each side of the regular hexagon has length \(r\) and a hexagon has 6 sides, the perimeter \(P\) of the hexagon is \(P = 6r\).

Step4: Calculate the circumference of the circle

The formula for the circumference of a circle is \(C = 2\pi r\).

Answer:

Length of segment \(AB\): \(r\)
\(\angle ACB\): \(60^{\circ}\)
Perimeter of the hexagon: \(6r\)
Circumference of the circle: \(2\pi r\)