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AMC10 2003b Test Paper
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Sample Question 23:

A regular octagon ABCDEFGHABCDEFGH has an area of one square unit. What is the area of the rectangle ABEFABEF?

(A) 122(B) 24(C) 21(D) 12(E) 1+24\textbf{(A) } 1 - \frac{\sqrt{2}}{2} \qquad\textbf{(B) } \frac{\sqrt{2}}{4} \qquad\textbf{(C) } \sqrt{2} - 1 \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{1+\sqrt{2}}{4}




Answer Keys

Question 23: D


Solutions

Question 23
Step 1: The area of a regular octagon is given by the formula ap2=1 \frac{ap}{2}=1 , where a a is the apothem and p p is the perimeter.

Step 2: This octagon's side length is p8 \frac{p}{8} and the apothem is 2a 2a .

Step 3: The area of the rectangle ABEF ABEF is given by the formula p8×2a=2ap8=ap4 \frac{p}{8} \times 2a = \frac{2ap}{8} = \frac{ap}{4} .

Step 4: From steps 1 and 3, we can observe that the area of the rectangle is half the area of the octagon.

Step 5: Since the area of the octagon is one unit square, the area of the rectangle ABEF ABEF is 12 \frac{1}{2} unit square.

Therefore, the answer is (D) 12\textbf{(D)}\ \frac{1}{2}.