Question

An equilateral triangle with side lengths equal to 12 StartRoot 3 EndRoot units is inscribed in a circle.

Half a side length of the equilateral triangle is 6 StartRoot 3 EndRoot units, so the apothem is
units long and the radius of the circle is
units long.

Each segment of the circle has an area equal to the difference between the areas of the sector and triangle, or (
π −
StartRoot 3 EndRoot) units2.

Answer 1

Half the side is 6√3, so the apothem (inradius) is 6 units and the circumradius is 12 units.

Each circular segment (for central angle 120° = 2π/3) has area
sector − triangle
= (1/3)π(12)^2 − (1/2)(12)^2 sin(120°)
= 48π − 36√3 units^2
(≈ 88.443 units^2).