Points $A(2,5)$ and $B(10,5)$ are the endpoints of a diameter of a circle graphed in a coordinate plane. How many square units are in the area of the circle? Express your answer in terms of $\pi$.
To find the area of the circle, I need to know its radius. The radius is half the length of the diameter, so I can use the distance formula to find the diameter. The distance between $A$ and $B$ is $\sqrt{(10-2)^2 + (5-5)^2} = \sqrt{64} = 8$ units. Therefore, the radius is half of that, or $4$ units. The area of a circle is $\pi$ times the radius squared, so the area of this circle is $\pi \cdot 4^2 = 16\pi$ square units. 