An Equilateral triangle (ABC) is inscribed inside a circle. The side lengths of the triangle are 8 cm. What is the radius of the circle?

Let the triangle be named ABC.

Name the centre of the circle O, and the mid-point of BC, D.
Triangle ODC is a right-triangle, right-angled at D.
radius, r = OC
and OD=r/2 (∠OCD=30°).
DC=8cm/2=4 cm.
So all the sides ΔODC are known in terms of r.
Use Pythagoras theorem to solve for r:
(4cm)²+(r/2)²=r²

To find the radius of the circle, we can use the relationship between the side length of an equilateral triangle and the radius of the circumscribed circle.

The radius of the circumscribed circle (R) is related to the side length (s) of an equilateral triangle by the formula:

R = (s / √3)

In this case, the side length of the equilateral triangle is 8 cm. Plugging this value into the formula, we have:

R = (8 cm / √3) ≈ 4.619 cm

Therefore, the radius of the circle is approximately 4.619 cm.

To find the radius of the circle, we can use the formula for the radius of a circumscribed circle of an equilateral triangle.

The formula states that the radius (r) of the circle is equal to the length of any side (s) of the equilateral triangle divided by the square root of 3.

In this case, the side length of the equilateral triangle is 8 cm. Thus, we can apply the formula:

r = s / √3

r = 8 cm / √3

To calculate the numerical value of the radius, we need to divide 8 cm by the square root of 3.