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# In the figure above, a circle with center O touches three sides of a

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Math Expert
Joined: 02 Sep 2009
Posts: 43810
In the figure above, a circle with center O touches three sides of a [#permalink]

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21 Nov 2017, 22:09
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In the figure above, a circle with center O touches three sides of a rectangle of length m and width n. Which of the following gives the area of the shaded region?

(A) πn^2/4
(B) mn – 2πn
(C) mn – πn^2
(D) mn – πn^2/2
(E) mn – πn^2/4

[Reveal] Spoiler:
Attachment:

2017-11-21_1031.png [ 3.95 KiB | Viewed 429 times ]
[Reveal] Spoiler: OA

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Re: In the figure above, a circle with center O touches three sides of a [#permalink]

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21 Nov 2017, 22:22
[(m*n)-{(3.14*(n^2)/4}]
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Joined: 22 May 2016
Posts: 1329
In the figure above, a circle with center O touches three sides of a [#permalink]

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26 Nov 2017, 20:02
Bunuel wrote:

In the figure above, a circle with center O touches three sides of a rectangle of length m and width n. Which of the following gives the area of the shaded region?

(A) πn^2/4
(B) mn – 2πn
(C) mn – πn^2
(D) mn – πn^2/2
(E) mn – πn^2/4

[Reveal] Spoiler:
Attachment:
2017-11-21_1031.png

Rectangle area = mn

The radius of the circle is $$\frac{1}{2}n$$

The area of the circle is
$$\pi r^2 = \pi* (\frac{1}{2}n)^2=\frac{n^2}{4}\pi$$

(Rectangle area) - (circle area) = Shaded area

Shaded area = $$mn - \frac{n^2}{4}\pi$$

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Re: In the figure above, a circle with center O touches three sides of a [#permalink]

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29 Nov 2017, 09:57
Bunuel wrote:

In the figure above, a circle with center O touches three sides of a rectangle of length m and width n. Which of the following gives the area of the shaded region?

(A) πn^2/4
(B) mn – 2πn
(C) mn – πn^2
(D) mn – πn^2/2
(E) mn – πn^2/4

[Reveal] Spoiler:
Attachment:
2017-11-21_1031.png

We see that the area of the shaded region is the difference between the area of the rectangle and the area of the circle. Since the width of the rectangle is n, we see that n is also the diameter of the circle. Thus, the radius of the circle is n/2.

Area of the rectangle = m x n = mn

Area of the circle = π x (n/2)^2 = π x (n^2/4) = πn^2/4

Thus, the area of the shaded region = mn - πn^2/4.

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Re: In the figure above, a circle with center O touches three sides of a   [#permalink] 29 Nov 2017, 09:57
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