The biggest rectangle in a semicircle

Question

A semicircle has radius 1. A rectangle is drawn inside it with its base lying along the diameter and its two upper corners on the arc.

What is the largest possible area of the rectangle, and what are its dimensions? Try to find the answer without calculus.

Accepted Answer

The maximum area is 1, for a rectangle of width root 2 and height 1 over root 2. Reflect. Mirror the semicircle and the rectangle across the diameter. You get a full circle of radius 1 with an inscribed rectangle of twice the area, whose corners all lie on the circle. Maximising the half-rectangle's area is the same as maximising the full rectangle's area. The inscribed rectangle. Any rectangle inscribed in a circle has the circle's diameter, 2, as its diagonal. If its sides are a and b, then a squared + b squared = 4. Its area is ab, and by the inequality of arithmetic and geometric means, a…

The biggest rectangle in a semicircle

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