How do you find the perimeter of a circle with triangle inside?

Answer 1

The same as a circle without a triangle inside: multiply twice the radius of the circle (which is its diameter) by π.

However, if you have been given some information about the triangle (such as its three lengths or two lengths and one angle, or one length and two angles, or one angle and the length of the side opposite it) which the circle circumscribes, then you can calculate twice the radius of the circle:

If you know one side (a) and the angle opposite it (a), then twice the radius of the circle can be found using the sine rule and is given by:

2R = a / sin A

→ circumference of the circle = aπ / sin A

If all you have is the lengths of the three sides, you can calculate the size of one of the angles A (with side a opposite it and sides b & c the other two sides) by the cosine rule and then use the above:

A = arc cos ((b² + c² - a²) / (2bc))

→ circumference = aπ / sin (arc cos ((b² + c² - a²) / (2bc)))