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

The apothem (x) is 11.62 units, the radius of the circle is 12 units, and the area of each segment is 48π − 23.25√3 square units.

What is a circle?

It is a locus of a point drawn equidistant from the center. The distance from the center to the circumference is called the radius of the circle.

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

And half a side length of the equilateral triangle is 6√3 units.

The apothem (x) is units long and the radius of the circle will be

x² = (9√3)² - (6√3)²

x² = 243 - 108

x² = 135

 x = 11.62

And the radius of the circle will be

$\begin{aligned} \cos 30 &= \dfrac{6\sqrt3}{r}\\\\\dfrac{\sqrt3}{2}&= \dfrac{6\sqrt3}{r}\\\\r &= 12 \end{aligned}$

Each segment of the circle has an area that will be

Area = the difference between the areas of the sector and triangle

Area = the areas of the sector - the areas of the triangle

The area of the sector will be

$\rm Sector's \ area= \dfrac{120}{360} \pi *12^2 \\\\Sector's \ area= 48\pi$

The area of the triangle will be

$\rm Triangle's \ area = \dfrac{1}{2*3} * 11.62 * 12\sqrt3\\\\Triangle's \ area = 23.25\sqrt3$

Then the area of each segment will be

Segment area = 48π − 23.25√3

More about the circle link is given below.

brainly.com/question/11833983