Find the length of a circle if the area of an inscribed regular triangle is 75√3.

Let ABC be a regular triangle (i.e. all its sides and angles are equal).
The area of ​​such a triangle is found by the formula S = (a ^ 2 * √3) / 4
We know the area: S = 75√3
Let’s substitute:
75√3 = (a ^ 2 * √3) / 4
75 = a ^ 2/4
a ^ 2 = 75 * 4
a ^ 2 = 300
a = ± 10√3
-10√3 extraneous root
Hence, side ABC = 10√3
Now let’s draw the height (median and bisector) of the VN in the triangle
NS = (10√3) / 2 = 5√3
By the Pythagorean theorem:
BH = √ ((10√3) ^ 2 – (5√3) ^ 2) = √ (100 * 3-25 * 3) = √ (300-75) = √225 = 15
point O is the center of the circle. And also this point lies on the VN.
According to the rule, the height is divided in the following proportion in a regular triangle: VO / OH = 2/1
It turns out that 1 part = 15/3 = 5
Then 2 parts (VO) = ​​5 * 2 = 10
a BO = circle radius (R)
The circumference is found by the formula: L = 2ПR
L = 2P * 10 = 20P
Answer: L = 20P