What is the area of the semicircle?

Question

A 1 inch by 2inch rectangle is inscribed inside a semicircle with the longer side on the diameter. What is the area of the semicircle?

solve without trigonometry if you can.

Answer 1

The midpoint of the diameter is the center of the circle. Draw a line segment from the midpoint of the diameter to the upper right corner of the rectangle. Since the rectangle is inscribed in the semicircle, the upper corners of the rectangle lie on the semicircle. The length of the line segment is a radius of the circle. The line segment is the hypotenuse of a right triangle with both legs of length 1.

So, by the Pythagorean Theorem, the radius of the circle has length √[(1² + 1²) = √2

Area of semicircle = (1/2)π(√2)² = π