Question

An equiangular octagon has four sides of length $1$ and four sides of length $\dfrac{\sqrt{2}}{2},$ arranged so that no two consecutive sides have the same length. What is the area of the octagon?

$\dfrac{7}{2}$

$\dfrac{7\sqrt{2}}{2}$

$\dfrac{5 + 4\sqrt{2}}{2}$

$\dfrac{4 + 5\sqrt{2}}{2}$

$7$

Solution:

Extend the four sides of length $1$ to form a square. Each short side $\dfrac{\sqrt{2}}{2}$ is the hypotenuse of an isosceles right triangle with legs $\dfrac{1}{2},$ and cutting these four corners from a square of side $1 + 2 \cdot \frac{1}{2} = 2$ gives the octagon. Its area is $2^2 - 4 \cdot \frac{1}{2}\left(\frac{1}{2}\right)^2 = 4 - \frac{1}{2} = \dfrac{7}{2}.$

Thus, the correct answer is A.

2005 AMC 10A Problem 20

Solution:

Problem 20 in Other Years