# Circles

Isaac Newton is one of the most famous scientists of all time, and His book Philosophiae Naturalis Principia Mathematica is arguably the most important book published in the history of science. He was a fascinating and enigmatic character. He suffered several major nervous breakdowns over the course of his life, and he spent many years researching alchemy (this investigation into the hidden forces of nature helped to lead him to his discovery of the action-at-a-distance nature of the law of gravity). You can read more about Newton here. He would make a great "excellent example" for your essay.

## 6/14 Geometry: Circles

In the figure above, a shaded circle with diameter  is tangent to a large semicircle with diameter  at points  and . Point  is the center of the semicircle, and  is perpendicular to . If the area of the large semicircle is 24, what is the area of the shaded circle?

As always, you should start by reading the problem carefully and identifying the bottom line. You are looking for the area of the shaded circle. Recall that you can calculate the area of a circle using the formula
$a=\pi r^2$

Now, you can assess your options. In this case, you need to find the value of r (the radius of the shaded circle) to calculate the area of the circle. You are given that the area of the large semicircle is 24, or in other words, the area of the circle (with A-B) as the diameter is 48. At this point you need to be very careful how you proceed. As long as you don't make any algebra errors, there are several ways to solve this problem, but some of them are much easier than others. You know that the area of the semicircle is 24 and you are looking for the area of the smaller circle. You also know that  R (the radius of the semicircle) is twice what r (the radius of the shaded circle) is. Now, you can write the following formulas

$\frac{\pi (2r)^2}{2}=\frac{4\pi r^2}{2}=2\pi r^2$

Note that all you have above is the area of the semicircle (which is the area of a circle with radius R=2r divided by 2). Since you know that the area of the semicircle is equal to 24 you could solve the formula for r and then calculate the area of the shaded circle. However, notice that you already have the area of the circle calculated above, it's just being multiplied by 2. So instead you can write the following

$2\pi r^2=24 \therefore \pi r^2 =12$

Remember to loop back and verify that you have answered the bottom line. Recall that at the beginning you defined r as the radius of the shaded circle and R as the radius of the semicircle. Now look at the answer choices and select the one that matches your answer.

(A) 8
(B) 10
(C) 12
(D) 14
(E) It cannot be determined from the information given.