Given a semicircle and a quarter of a circle with equal perimeters, which one will have a greater area?

Let P = Perimeter and A = Area. For a quarter circle

For a semicircle,

So,

and .

.

Since the perimeters are equal, we can compare the denominators of the two areas. Obviously, the denominator of A2, the semicircle is larger. So, the area of the quarter circle will be larger if the semicircle and quarter circle have equal perimeters.

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