2005 AMC 10A Problem 20

Every interior angle of an equiangular octagon is $135^\circ;$ enclose the octagon in a larger square

The four short sides are hypotenuses of isosceles right triangles with legs $\frac{1}{2},$ cut from a square of side $2$

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.