A hexagon has a radius of 23 cm

∠ACB=60°, AC=r, Perimeter=6r units, Circumference=6.28r≈6r

Step-by-step explanation:

Given a regular hexagon inscribed in circle C with radius r. Also, ∠CAB = 60°.  

In ΔACB,

∠CAB = 60° and also ∠CAB=∠CBA as CA=CB (both radii of same circle) ∴ ∠CBA=60°

By angle sum property of triangle

∠CAB+∠CBA+∠ACB=180°

⇒ 60°+60°+∠ACB=180°

⇒ ∠ACB=60°

All angles of ΔABC are equal which implies ABC is an equilateral triangle.

∴ AB=BC=AC=r

Perimeter of hexagon=6(Length of one side of hexagon)

                                     =6(AB)=6r units

Circumference of circle==2(3.14)r=6.28r≈6r units

∴Perimeter of hexagon is equal to circumference of circle.

Circumference of circle is =2(3.14)r=6.28r units

Hence, ∠ACB=60°, AC=r, Perimeter=6r units, Circumference=6.28r≈6r