MATH ITEM:

The figure above shows the graph of a circle with center O. If PQRO is a rectangle with area 12, what is the area of Circle O?

(A)     3
(B)     4
(C)     16
(D)     25
(E)     Cannot be determined.

Correct Answer: (D)

MATH EXPLANATION:
First, you will probably want to review the “official” method for solving the item with your students. Since the area of rectangle OPQR is 12:

At this point, you might recognize a 3-4-5 triangle. If not, you can use the Pythagorean Formula to find the length of :

Then, since is a radius of the circle, you can find the area of the circle:

The “official” approach requires the use of several concepts and the ability to integrate them into a coherent solution. Many students may find it easier to use an alternative strategy.

Students can use the figure itself to find the answer: Simply measure the radius of the circle. The use of any straight edge, such as the edge of the answer sheet, will make this measurement possible. Mark the length of the radius on the edge of the answer sheet (using a light pencil mark that will be erased once the problem is solved):

Then, compare that length with the known length of 4. It appears that the radius of the circle is about 1/4 more than the length of the rectangle, or about 5 units:

Only the final step of the calculation must then be applied:

Another alternative strategy seeks to estimate the area of the circle. The rectangle with area 12 seems to be about 2/3 the size of the quarter of the circle in the first quadrant. So, the portion of the circle in that quadrant has an area of approximately 18. . Which answer choice is close to 72? Since , divide by 3. Answer choice (D) is the closest: . Though this estimate is rough, it is sufficiently precise to establish that (D) must be the correct answer choice.