Question

isla says the areas of the two shaded regions are equal. clara says the shaded region in the first figure has a larger area. who is correct? isla is correct because each figure subtracts an equal area from the area of the square. isla is correct because it is the sum of the area of the circle and the area of the square. clara is correct because the area of the circle is less than that of the two semicircles. clara is correct because the area of the two semicircles is less than that of the circle.

Explanation:

Step1: Let the side - length of the square be \(s\) and the radius of the circle be \(r\). In both figures, assume the circle(s) is(are) inscribed in the square(s). For the first figure with a single - circle, the diameter of the circle \(d = s\) (since the circle is inscribed in the square), so \(r=\frac{s}{2}\). The area of the square \(A_{square1}=s^{2}\), and the area of the circle \(A_{circle1}=\pi r^{2}=\pi(\frac{s}{2})^{2}=\frac{\pi s^{2}}{4}\). The area of the shaded region \(A_{shaded1}=s^{2}-\frac{\pi s^{2}}{4}=s^{2}(1 - \frac{\pi}{4})\).

Step2: For the second figure with two semi - circles, the diameter of each semi - circle is also \(s/2\), and the radius of each semi - circle is \(r=\frac{s}{4}\). The combined area of the two semi - circles is equivalent to the area of one full - circle with radius \(r = \frac{s}{2}\) (because the sum of the areas of two semi - circles with the same radius is the area of a full - circle). The area of the square \(A_{square2}=s^{2}\), and the area of the two semi - circles (equivalent to one full - circle) \(A_{circle2}=\pi r^{2}=\pi(\frac{s}{2})^{2}=\frac{\pi s^{2}}{4}\). The area of the shaded region \(A_{shaded2}=s^{2}-\frac{\pi s^{2}}{4}=s^{2}(1 - \frac{\pi}{4})\).

Answer:

Isla is correct because each figure subtracts an equal area from the area of the square.