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# In the figure above, AC = BC = 8, angle C = 90°, and the circular arc

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Math Expert
Joined: 02 Sep 2009
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In the figure above, AC = BC = 8, angle C = 90°, and the circular arc  [#permalink]

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10 Mar 2015, 06:07
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In the figure above, AC = BC = 8, angle C = 90°, and the circular arc has its center at point C. Find the area of the shaded region.

A. $$8\pi-32$$
B. $$16\pi-32$$
C. $$16\pi-64$$
D. $$32\pi-32$$
E. $$32\pi-64$$

Kudos for a correct solution.

Attachment:

gm-tuaaof_img1.png [ 5.79 KiB | Viewed 2446 times ]

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Re: In the figure above, AC = BC = 8, angle C = 90°, and the circular arc  [#permalink]

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10 Mar 2015, 06:33
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Hi

Area of sector = (90/360)*pi(8^2) = 16pi

Area of triangle = (1/2)*8*8 = 32

Area of shaded region = area of sector - area of triangle = 16pi-32
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Re: In the figure above, AC = BC = 8, angle C = 90°, and the circular arc  [#permalink]

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10 Mar 2015, 19:58
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Area of shaded region = (area of circle/4) - area of triangle = (pi* 8^2)/4 - (8*8/4) = 16 pi - 32

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Re: In the figure above, AC = BC = 8, angle C = 90°, and the circular arc  [#permalink]

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15 Mar 2015, 21:11
Bunuel wrote:

In the figure above, AC = BC = 8, angle C = 90°, and the circular arc has its center at point C. Find the area of the shaded region.

A. $$8\pi-32$$
B. $$16\pi-32$$
C. $$16\pi-64$$
D. $$32\pi-32$$
E. $$32\pi-64$$

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

There’s not a single formula we can use to get the answer, but by combining a few formulas, we can calculate this.

First, think about the circle. The circle has radius r = 8, so its total area would be $$A=\pi{r^2}=64\pi$$.
gm-tuaaof_img5

This entire figure is a quarter of the circle, so that area would be $$quarter \ circle=16\pi$$.

Now, the shaded area (technically known as a circular segment), would have an area of
(circular segment) = (quarter circle) – (triangle ABC)

Well, we already have the area of the quarter circle. The triangle would have an area of (1/2)bh = 32. Therefore, the area of the segment is $$area=16\pi-32$$.

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Re: In the figure above, AC = BC = 8, angle C = 90°, and the circular arc  [#permalink]

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27 Feb 2018, 10:24
Top Contributor
Bunuel wrote:
Attachment:
gm-tuaaof_img1.png
In the figure above, AC = BC = 8, angle C = 90°, and the circular arc has its center at point C. Find the area of the shaded region.

A. $$8\pi-32$$
B. $$16\pi-32$$
C. $$16\pi-64$$
D. $$32\pi-32$$
E. $$32\pi-64$$

Kudos for a correct solution.

Area of shaded region = (area of sector) - (area of triangle)

Area of triangle = (base)(height)/2

The sector ABC is 1/4 of a circle of radius 8
So, area of sector = (1/4)(π)(8²)
= 16π

area of triangle = (8)(8)/2
= 32

So, area of shaded region = (16π) - (32)

Cheers,
Brent
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Re: In the figure above, AC = BC = 8, angle C = 90°, and the circular arc   [#permalink] 27 Feb 2018, 10:24
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