Question

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

$\dfrac{\sqrt{2}-1}{2}$

$\dfrac{1}{4}$

$\dfrac{2-\sqrt{2}}{2}$

$\dfrac{\sqrt{2}}{4}$

$2-\sqrt{2}$

Solution:

Assume the octagon has edge length $1.$ The four corner triangles are right isosceles with legs $\dfrac{\sqrt2}{2}$ and area $\dfrac14$ each. The four rectangles are $1$ by $\dfrac{\sqrt2}{2}$ with area $\dfrac{\sqrt2}{2}$ each, and the center square has area $1.$

The total area is $4\cdot\dfrac14+4\cdot\dfrac{\sqrt2}{2}+1=2+2\sqrt2.$ The probability of hitting the center square is $\dfrac{1}{2+2\sqrt2}=\dfrac{\sqrt2-1}{2}.$

Thus, the correct answer is A.

2011 AMC 12B Problem 12

Solution:

Problem 12 in Other Years