Problem Definition

In the diagram below, there is a rectangle, a quarter circle, and three semicircles. If the rectangle has an area of 4 cm², what is the area of the shaded region between the semicircles?

Solution

This problem showcases the interplay between circular and rectangular geometry. The challenge combines three semicircles with a rectangular boundary, creating a visually intriguing configuration where the shaded green region has a beautifully clean answer.

Understanding the Configuration

Consider a rectangle with horizontal dimension $x$ and vertical dimension $y$. The geometric arrangement consists of:

• One large semicircle with diameter $x + y$, arching over the entire top
• One medium semicircle with diameter $x$, sitting along the left portion of the base
• One small semicircle with diameter $y$, sitting along the right portion of the base

Mathematical Foundation

The rectangle has horizontal length $x$ and vertical length $y$, so its area gives us the core constraint:

$$xy = 4$$

Because the large semicircle spans the full base of $x + y$, the quarter-circle on the right has radius $y$, and therefore the small semicircle along the right portion of the diameter has diameter $y$. The three semicircles have diameters:

• Large semicircle: diameter $x + y$, radius $\frac{x+y}{2}$
• Medium semicircle: diameter $x$, radius $\frac{x}{2}$
• Small semicircle: diameter $y$, radius $\frac{y}{2}$

Computing Individual Areas

The area of each semicircle is $\frac{1}{2}\pi r^2$:

$$A_{\text{large}} = \frac{\pi}{2}\left(\frac{x+y}{2}\right)^2 = \frac{\pi(x+y)^2}{8}$$ $$A_{\text{medium}} = \frac{\pi}{2}\left(\frac{x}{2}\right)^2 = \frac{\pi x^2}{8}$$ $$A_{\text{small}} = \frac{\pi}{2}\left(\frac{y}{2}\right)^2 = \frac{\pi y^2}{8}$$

Determining the Shaded Area

The green shaded region is the large semicircle minus the medium and small semicircles:

$$A_{\text{shaded}} = A_{\text{large}} - A_{\text{medium}} - A_{\text{small}}$$ $$A_{\text{shaded}} = \frac{\pi(x+y)^2}{8} - \frac{\pi x^2}{8} - \frac{\pi y^2}{8}$$ $$A_{\text{shaded}} = \frac{\pi}{8}\left[(x+y)^2 - x^2 - y^2\right]$$ $$A_{\text{shaded}} = \frac{\pi}{8}\left[x^2 + 2xy + y^2 - x^2 - y^2\right]$$ $$A_{\text{shaded}} = \frac{\pi}{8}(2xy) = \frac{\pi}{4}xy$$

The Elegant Solution

Substituting the rectangle constraint $xy = 4$:

$$A_{\text{shaded}} = \frac{\pi}{4} \times 4 = \pi$$

The shaded area is exactly $\pi \text{ cm}^2$, regardless of the specific dimensions of the rectangle - only its area matters. This is a beautifully elegant result: the cross terms from expanding $(x+y)^2$ produce exactly $2xy$, which cancels cleanly with the constraint, leaving $\pi$ as the answer.