The perimeter of a regular hexagon inscribed in a circle is 30cm. What is the area of a triangle inscribed

The perimeter of a regular hexagon inscribed in a circle is 30cm. What is the area of a triangle inscribed in the same circle?

Let’s find the length of one side of our hexagon:

a = P / 6 = 30/6 = 5 cm.

If you draw the radii to all the vertices of a regular hexagon, it turns out that the angle between two adjacent radii is the same and is equal to:

360º / 6 = 60º.

Consider a triangle formed by two radii and a side of a hexagon. It is isosceles, since both radii are equal. Let’s find what the angles at its base are equal to:

(180 – 60) / 2 = 120/2 = 60º.

This means that all angles of the triangle are equal and it is equilateral. That is, the radius of the circumscribed circle is 5 m. The formula for finding the area of ​​a triangle inscribed in a circle is:

S = abc / 4R, where a, b and c are the sides of the triangle, R is the radius of the circle.

Hence, in this case, this formula will look like this:

S = abc / (4 * 5) = abc / 20.

Answer: the area of ​​a triangle inscribed in a circle is abc / 20 m².