# Circles

Anytime something new happens to something very old, the result is a rich current event that could be interpreted in many different ways.  The Catholic Church has chosen a new pope, and for the first time ever, the pope is from the Americas.  Look for broad themes in this article that would make it easy to relate this current event to an SAT essay topic.

## Geometry: Circles

Read the following SAT test question and then select the correct answer.

Read the question carefully and identify the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that your answer matches the bottom line.
In the figure above, a shaded circle with diameter  is tangent to a large semicircle with diameter  at points C and D. 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?

Bottom Line: A sm =? (What is the area of the small, shaded circle?)

Assess your Options:  There are two good ways to approach this problem.  Both ways require you to know the formula for the area of a circle. On collegeboard.org you will find a method that is especially efficient for students who are good at writing equations.  The method used here will focus on geometry skills and estimation in order to avoid the mistakes that often come with working more abstract formulas.

Attack the Problem:  You know the most about the large circle, so start there.  A semicircle is just half of a whole circle.  Therefore, to find the area of the whole circle, you would simply double the 24.

24 × 2 = 48

If you know the area of the large circle, you can use the area formula to find out more information.  The area of a circle is $A=\Pi(r)^{2}$  Plug in the area you just found to find the radius.

Note: working backwards using the area formula for a circle is difficult, because using pi will always result in icky decimals.  If you glance at your answer choices, all of them are whole numbers.  You can estimate pi as 3 instead of 3.14 in order to keep this problem as easy as possible.

48 = 3r²
16 =
4 = r

You now have the radius for the big circle.  Now look back up at the diagram.  The radius for the big circle is also the diameter for the little circle!  If the diameter of the little circle is 4, the radius will be half of that.  Once you know that the radius of the little circle is 2, you are ready to find the area!

$A=\Pi(r)^{2}$
A = 3 × 2²

A = 3 × 4
A = 12

Loop Back:  You found the area of the small circle, so you are ready to look at your answer choices.

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