Question

sla 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.

Answer 1

Isla is correct — each shaded area equals the area of the square minus the area of a circle of diameter equal to the square’s side.

Let the square side be s. In the first figure the shaded area = s^2 − π(s/2)^2 = s^2(1 − π/4).
In the second figure the two semicircles together have area π(s/2)^2 (they make a full circle), so the shaded area = s^2 − π(s/2)^2 = s^2(1 − π/4) as well.

So the two shaded regions are equal (choice 1).