Find the area of a circle described about a triangle with sides 7 cm 8 cm 9 cm.

Find the area S of this triangle. To do this, we will use Heron’s formula:
S = √ (p * (p – a) * (p – b) * (p – c)), where a, b and c are the sides of the triangle, and p is the half perimeter of the triangle, that is, half the sum of the sides of the triangle:
p = (a + b + c) / 2.
By the condition of the problem, a = 7, b = 8, c = 9, therefore the semiperimeter p of this triangle is equal to:
p = (7 + 8 + 9) / 2 = 24/2 = 12,
and the area of ​​this triangle is:
S = √ (p * (p – a) * (p – b) * (p – c)) = S = √ (12 * (12 – 7) * (12 – 8) * (12 – 9)) = √ (12 * 5 * 4 * 3) = √ (12 * 5 * 12) = 12√5.

Determine the radius R of the circumscribed circle using the formula R = a * b * c / (4 * S):
R = 7 * 8 * 9 / (4 * 12√5) = 21 / (2√5).

Now, using the formula S = π * R ^ 2, we find the area of ​​the circumscribed circle:
π * R ^ 2 = π * (21 / (2√5)) ^ 2 = π * (21) ^ 2 / (2√5)) ^ 2 = π * 441/20 = π * 22.05.

Answer: the area of ​​a circle circumscribed around this triangle is π * 22.05.