A triangle inscribed in a circle with the diameter as its hypotenuse is a right triangle. We can use the Pythagorean theorem to find the length of AC.

Let x = the length of AC / the circle's diameter
\(10^2 + 24^2 = x^2\)
\(676 = x^2\)
\(x = 26\)

Now we know the radius of the circle is 13 and we can find the area of the semicircle:
\(\frac{1}{2} * \pi * 13^2 = 84.5 \pi\)

Find the area of the triangle:
\(\frac{1}{2} * 10 * 24 = 120\)

Subtract the area of the triangle from the area of the semicircle to get the answer:
\(84.5 \pi - 120\)