The sides of the triangle are 4cm, 13cm and 15cm. Calculate the radius of the circle around the triangle.

The radius of a circle circumscribed about a triangle can be found by the formula:
R = abc / 4S,
where R is the radius of the circumscribed circle, a, b and c are the sides of the triangle, S is the area of the triangle.
The area of an arbitrary triangle, for which all three sides are known, can be found by Heron’s formula:
S = √ (p (p – a) (p – b) (p – c)),
where p is the semiperimeter of the triangle.
p = (a + b + c) / 2;
p = (4 + 13 + 15) / 2 = 16 (cm).
S = √ (16 (16 – 4) (16 – 13) (16 – 15)) = √ (16 * 12 * 3 * 1) = √576 = 24 (cm square).
Substitute the known values into the formula and find the radius of the circumscribed circle:
R = 4 * 13 * 15/4 * 24 = 780/96 = 8.125 (cm).
Answer: R = 8.125 cm.