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# Two semi-circles are drawn on adjacent sides of a square with side len

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Math Expert
Joined: 02 Sep 2009
Posts: 65764
Two semi-circles are drawn on adjacent sides of a square with side len  [#permalink]

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16 Jun 2020, 10:05
00:00

Difficulty:

35% (medium)

Question Stats:

81% (01:43) correct 19% (03:36) wrong based on 21 sessions

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Two semi-circles are drawn on adjacent sides of a square with side length 1 as shown above. What is the area of the shaded region?

A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4

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2020-06-16_2204.png [ 41.75 KiB | Viewed 418 times ]

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Re: Two semi-circles are drawn on adjacent sides of a square with side len  [#permalink]

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16 Jun 2020, 10:15
4
Bunuel wrote:

Two semi-circles are drawn on adjacent sides of a square with side length 1 as shown above. What is the area of the shaded region?

A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4

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The attachment 2020-06-16_2204.png is no longer available

Attachment:

Screenshot 2020-06-16 at 11.43.06 PM.png [ 36.51 KiB | Viewed 385 times ]

Area of shaded region = Area of half of square - 2x + 2x = Area of half of square = 1/2

IMO C
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Kinshook Chaturvedi
Email: kinshook.chaturvedi@gmail.com
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Joined: 16 May 2011
Posts: 283
Concentration: Finance, Real Estate
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Two semi-circles are drawn on adjacent sides of a square with side len  [#permalink]

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16 Jun 2020, 13:00
1
Kinshook
great visualisation.
i was fighting to calculate this little leaf . now i see it just comletes the triangle to half of the circrle.
i am with you: 1/2

is there a mathematical way to calculate the "leaf" area?
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Two semi-circles are drawn on adjacent sides of a square with side len  [#permalink]

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20 Jun 2020, 12:42
Bunuel wrote:

Two semi-circles are drawn on adjacent sides of a square with side length 1 as shown above. What is the area of the shaded region?

A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4

Solution:

First, draw a diagonal from the upper left to the lower right of the square, creating two isosceles right triangles. Note that the two white regions in the upper right triangle, if they are combined, are identical to the shaded red lens-shaped figure in the lower left triangle. Thus, the shaded region is exactly half of the area of the square. Since a side of the square is 1, the area of the square is 1, and thus, the area of the shaded region is ½.

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Two semi-circles are drawn on adjacent sides of a square with side len   [#permalink] 20 Jun 2020, 12:42

# Two semi-circles are drawn on adjacent sides of a square with side len

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