Geometry: shaded region formed by smaller circle being removed from larger circle; area is 64pi; smaller radius is 6

amp1053

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Feb 27, 2019
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I’m really unsure of how to do this problem and need some help. The image shows a large shaded circle with a smaller unshaded circle inside. It reads, "This figure consists of 2 concentric circles. If the shaded area is 64 (pi) sq. in. and the smaller circle has a radius of 6 in., what is the radius, in inches of the larger circle?" I wasn’t at school today and I have a little work done (as shown in the photo), but I’m really struggling and I have no clue what to do. Thanks for the help!! :)
 

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Dr.Peterson

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You've found that the inner circle has an area of \(\displaystyle 36\pi\) sq. in. What is the area of the outer circle (including shaded and unshaded parts)? What radius would give that area? (Note that it's better not to calculate decimal values for a problem like this, since they gave you the shaded area in terms of pi.)
 

amp1053

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So I added the areas to get a total area of 100 sq. in., but I am lost on how to find a radius for this area. I’m sorry that I’m asking so many questions.
 

Dr.Peterson

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Learning math is all about asking questions! (Sometimes you ask others, sometimes you ask yourself.)

First, don't forget the pi! You're adding \(\displaystyle 36\pi + 64\pi = 100\pi\), not just 100.

Suppose I gave you this problem:

If a circle's area is \(\displaystyle 100\pi\) sq. in., what is its radius?​

You would, I hope, write an equation that expresses this fact, and try to solve it. What is the formula for the area of a circle?
 

Harry_the_cat

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Almost. \(\displaystyle 36\pi +64\pi = 100 \pi\) square inches.

Next step: Write down the formula for the area of a circle in terms of r. Let this equal \(\displaystyle 100\pi\) and solve for r.
 

Harry_the_cat

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Snap, Dr P.
 
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