Circles and Triangles

Geometry: Circles and Triangles

Read the following test question and then select your answer.

When you read, make sure you read carefully so that you don't miss anything important. Write down the bottom line, then assess your options and attack the problem with the most efficient method you know. Finally, loop back to make sure you answered the right question.



The circle shown above has center O and a radius of length 5. If the area of the shaded region is  , what is the value of x?

If this problem seems impossible at first glance, don't panic. It will have several steps, but it is far from impossible. Keep in mind that you don't need to know the entire path to the right answer when you start working, and in this problem that would be incredibly difficult. Just follow the steps of the Knowsys Method.

Before you start, notice that the picture says "not drawn to scale." That means that the test makers deliberately distorted it so it wouldn't help you as much, but you can still get some useful facts out of it. For example, O is both the center of the circle and one corner of the triangle. The fact that it is a right triangle is also likely to prove useful.


First, write down the bottom line.



Next, assess your options. When I ask my students what they could do when facing a problem like this, sometimes their answer is, "Cry." You could, if it would make you feel better, but on the test that will cost you time, and during practice it won't make the problem go away. So what do you do next?

Look at what information the problem gives you. You have the radius of the circle and the area of part of the circle. You can use the radius to find the total area...







...and then compare the two amounts.





Now you've figured out that the shaded area is four-fifths of the total area of the circle. What can you do with that? Well, the remaining fifth of the circle is within the triangle, which means that the corner with its vertex at O has a fifth of the degrees around the center of the circle.



So that angle measures 72 degrees. Since this is a right angle, it is now easy to calculate x.



Glance up to the bottom line to make sure you solved what you needed to. Then, look at the answer choices and select the right one.

A) 18

B) 36

C) 45

D) 54

E) 72


The answer is A.

On sat.collegeboard.org, 33% of responses were correct.



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