Area & perimeter

The area of any triangle is half its base times its height: \(A=\frac12 bh\). A diagonal connects two non-adjacent vertices of a polygon. From each vertex you can draw a diagonal to every other vertex except itself and its two neighbours, giving \(n-3\) diagonals per vertex. With \(n\) vertices this counts each diagonal twice, so the total is \(\frac{n(n-3)}2\). For a grid, a \(k\times k\) square can start in \((9-k)^2\) positions on an \(8\times8\) board, so the total number of squares is \(1^2+2^2+\cdots+8^2\).