The area of a regular triangle is 12√3. Find the area of the circle around this triangle.

In a regular triangle, all sides and angles are equal, which means that each of the angles is 180 ° / 3 = 60 °.

The area of ​​a triangle can be found as half the product of two adjacent sides by the sine of the angle between them:

S = a ^ 2 * sin α / 2.

From here, knowing the area, we can find the side:

a2 = 2 * S / sin α = 2 * 12√3 / sin 60 ° = 24√3 / (√3 / 2) = 48;

a = √48 = 4√3 – side of a given regular triangle.

The radius of a circle circumscribed about a regular triangle is determined by the formula:

R = a / √3 = 4√3 / √3 = 4.

Scircle = n * R ^ 2 = n * 4 ^ 2 = 16n ≈ 50.26 – the area of ​​a circle circumscribed about a given regular triangle.