# Answer to Question #20742 in Abstract Algebra for daksh

Question #20742

semi circulat region and square region have equal perimeter area of a square exceeds area of semicircular by 7 cm find the perimeter and area

Expert's answer

Let a be side of square. Then square perimeter is 4a and square area is a^2Let r be radius of semicircle. Semicircular perimeter then is pi*r and semicircular area ispi*r^2 / 2

So we have4a=pi*r+2r = r(pi+2) (1)

a^2 = pi*r^2 / 2 + 7 (2)

from (1) we find

a = (pi+2)*r/4

substituting to (2) gives

(pi+2)^2 r^2 / 16 = pi* r^2 / 2 + 7

(pi+2)^2 / 8 * r^2 = pi * r^2 + 14

((pi^2+4pi+4)/8 - pi) r^2 = 14

0.1629 r^2 = 14

r = 9.27

hence

perimeter of semicircle and square is 47.66

area of square is 141.97

are of circle is 134.97

So we have4a=pi*r+2r = r(pi+2) (1)

a^2 = pi*r^2 / 2 + 7 (2)

from (1) we find

a = (pi+2)*r/4

substituting to (2) gives

(pi+2)^2 r^2 / 16 = pi* r^2 / 2 + 7

(pi+2)^2 / 8 * r^2 = pi * r^2 + 14

((pi^2+4pi+4)/8 - pi) r^2 = 14

0.1629 r^2 = 14

r = 9.27

hence

perimeter of semicircle and square is 47.66

area of square is 141.97

are of circle is 134.97

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