In an isosceles triangle, the radius of the circumscribed circle is 5 cm, and the base is 8 cm, find the area of the triangle.

Since the ABC triangle is isosceles, the center of the circumscribed circle about the triangle lies at the height and median BН, drawn to the base of the AC.

ОА = ОВ = R = 5 cm.

AH = CH = AC / 2 = 8/2 = 4 cm.

In a right-angled triangle AOН, according to the Pythagorean theorem, we determine the length of the leg OH.

OH ^ 2 = AO ^ 2 – AH ^ 2 = 25 – 16 = 9.

OH = 3 cm.

Then the length of the height BH = OB + OH = R + OH = 5 + 3 = 8 cm.

Let’s define the area of the triangle ABC.

Savs = AC * ВН / 2 = 8 * 8/2 = 64/2 = 32 cm2.

Answer: The area of triangle ABC is 32 cm2.