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805+ (Hard)|   Geometry|      
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TheNightKing
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The perimeter of a square is equal to twice the perimeter of 28 Jul 2020, 16:47
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The perimeter of a square is equal to twice the perimeter of a rectangle of length of 8cm and breadth 7 cm. What is the circumference of a semicircle whose diameter is equal to the side of the square? (Rounded off to the two decimal places)

(A) 38.57 cms.
(B) 23.57 cms.
(C) 42.46 cms.
(D) 47.47 cms.
(E) 35.87 cms.­
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The perimeter of a square is equal to twice the perimeter of Updated on: 29 Jul 2020, 06:51
Originally posted by CrackverbalGMAT on 28 Jul 2020, 20:07.
Last edited by CrackverbalGMAT on 29 Jul 2020, 06:51, edited 1 time in total.
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TheNightKing
The perimeter of a square is equal to twice the perimeter of a rectangle of length of 8cm and breadth 7 cm. What is the circumference of a semicircle whose diameter is equal to the side of the square? (Rounded off to the two decimal places)
(a) 38.57 cms.
(b) 23.57 cms.
(c) 42.46 cms.
(d) 47.47 cms.
(e) 35.87 cms.
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The perimeter of a square = 4a

The perimeter of a rectangle = 2(l + b)

Given l = 8cm and b = 7 cm.

Perimeter = 2(8 + 7) = 30

Given perimeter of the square = 2 * perimeter of the rectangle

4a = 2 * 30 = 60

a = 15 cm

Diameter of the semicircle = Side of the square = 15 cm

Radius = \(\frac{Diameter}{2} \)= \(\frac{15}{2}\) cm

Circumference of the semicircle = \(\frac{1}{2}\) * 2 \(\pi\) r + Diameter = \(\pi\) r + d = \(\frac{22}{7}\) * \(\frac{15}{2 }\) + 15 = 23.57 + 15 = 38.57 cm

Option A

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The perimeter of a square is equal to twice the perimeter of 28 Jul 2020, 20:32
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CrackVerbalGMAT
TheNightKing
The perimeter of a square is equal to twice the perimeter of a rectangle of length of 8cm and breadth 7 cm. What is the circumference of a semicircle whose diameter is equal to the side of the square? (Rounded off to the two decimal places)
(a) 38.57 cms.
(b) 23.57 cms.
(c) 42.46 cms.
(d) 47.47 cms.
(e) 35.87 cms.

The perimeter of a square = 4a

The perimeter of a rectangle = 2(l + b)

Given l = 8cm and b = 7 cm.

Perimeter = 2(8 + 7) = 30

Given perimeter of the square = 2 * perimeter of the rectangle

4a = 2 * 30 = 60

a = 15 cm

Diameter of the semicircle = Side of the square = 15 cm

Radius = \(\frac{Diameter}{2} \)= \(\frac{15}{2}\) cm

Circumference of the semicircle = \(\frac{1}{2}\) * 2 \(\pi\) r = \(\pi\) r = \(\frac{22}{7}\) * \(\frac{15}{2 }\)= 23.57 cm

Option B

Arun Kumar
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Hi Arun,

Don't we need to add the diameter while calculating the circumference of a semi-circle?
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The perimeter of a square is equal to twice the perimeter of 29 Jul 2020, 06:43
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nkme2007

Don't we need to add the diameter while calculating the circumference of a semi-circle?
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Yes. The perimeter of our rectangle is 30, so the perimeter of the square is 60, and one side of the square is 15. That's the diameter of the semicircle. So the length of the curved part of the semicircle is half the circumference of a circle with diameter 15, and is thus (1/2)(15π) = 15π/2, and adding the diameter to that, the answer is 15 + 15π/2.

I'd personally just use the estimate π ~ 3 now, from which we see the answer is roughly 37.5 (but is slightly larger than that), and so A must be right. You could also notice that π < 3.2, so 15 + 15π/2 is less than 15 + 24 = 39, if you wanted to be sure the answer is not C or D.
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The perimeter of a square is equal to twice the perimeter of 29 Jul 2020, 19:48
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IanStewart
nkme2007

Don't we need to add the diameter while calculating the circumference of a semi-circle?

Yes. The perimeter of our rectangle is 30, so the perimeter of the square is 60, and one side of the square is 15. That's the diameter of the semicircle. So the length of the curved part of the semicircle is half the circumference of a circle with diameter 15, and is thus (1/2)(15π) = 15π/2, and adding the diameter to that, the answer is 15 + 15π/2.

I'd personally just use the estimate π ~ 3 now, from which we see the answer is roughly 37.5 (but is slightly larger than that), and so A must be right. You could also notice that π < 3.2, so 15 + 15π/2 is less than 15 + 24 = 39, if you wanted to be sure the answer is not C or D.
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Thank You.IanStewart
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