Problem: Given a semicirle and a quarter circle with equal perimeter, which has the largest area?
We begin with the quarter circle. We know that the perimeter and area of the quarter circle for some radius r1 is :
Solving for r1, we have:
Similarly, we can find the perimeter and area of the semicircle for some r2.
Solving for r2:
Using substitution we can now compare the two areas.
A1 represents the area of the quarter circle and A2 represents the area of the semicircle.
Since we know that the perimeter P is equal for both sectors, we can simply compare the denominators of the two. It is clear that the denominator of the semicirlce is larger than the denominator of the quarter circle. From this we may conclude that when the two sectors have equal perimeter, the quarter circle has more area.