### Instruction: A circle is circumscribed about an equilateral triangle with side lengths of $9$ units each.  What is the area of the circle, in square units? Express your answer in terms of $\pi$.### Response: To find the area of the circle, I need its radius.
I recall that the radius of a circumscribed circle is equal to two-thirds of the height of the triangle.
To find the height of the triangle, I can use the Pythagorean theorem on one of the right triangles formed by an altitude and a half-side.
The altitude splits the base into two segments of $4.5$ units each, and the hypotenuse is $9$ units.
So the height squared is $9^2 - 4.5^2 = 56.25$.
Taking the square root, I get that the height is $7.5$ units.
Therefore, the radius of the circle is two-thirds of that, or $5$ units.
The area of the circle is then $\pi r^2 = 25\pi$ square units.