Question

comparing the areas of two composite figures
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:

To determine who is correct, we analyze the area of the shaded regions. Both figures are squares with a circular (or semicircular) unshaded area. In the first figure, the unshaded area is a full circle. In the second figure, the two semicircles combine to form a full circle (since the sum of two semicircles with the same diameter as the circle in the first figure is equal to the area of that circle). The shaded area in each case is the area of the square minus the area of the unshaded circular region. So, the shaded areas should be equal because each subtracts an equal (circular) area from the square's area.

• The second option is incorrect because the shaded area is square area minus circle area, not sum.
• The third and fourth options are incorrect because the area of two semicircles (with diameter equal to the square's side) is equal to the area of one full circle (since the area of a semicircle is $\frac{1}{2}\pi r^2$, two semicircles would be $2\times\frac{1}{2}\pi r^2=\pi r^2$, same as a full circle with radius $r$ (where the diameter of the circle is equal to the square's side, so radius is half the side)).

So, Isla is correct for the reason stated in the first option.

Answer:

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