2002 AMC 10B 第 17 题

Place the octagon on a grid; each slanted side moves $\sqrt2$ horizontally and vertically

$DG$ is a horizontal chord; use it as the base and the vertical distance from $A$ as the height

Set the octagon on coordinate axes with the axis-aligned sides of length $2$ and each slanted side spanning $\sqrt2$ horizontally and $\sqrt2$ vertically. Then $$\begin{aligned} A &= (\sqrt2, 0), \\ D &= (2 + 2\sqrt2, 2 + \sqrt2), \\ G &= (0, 2 + \sqrt2). \end{aligned}$$

Since $D$ and $G$ share the height $2 + \sqrt2,$ segment $DG$ is horizontal with length $2 + 2\sqrt2,$ and the height from $A$ up to that level is $2 + \sqrt2.$

Therefore $$\begin{aligned} [\triangle ADG] &= \tfrac12(2 + 2\sqrt2)(2 + \sqrt2) \\ &= (1 + \sqrt2)(2 + \sqrt2) \\ &= 4 + 3\sqrt2. \end{aligned}$$

Thus, the correct answer is C.