Mathematical Enigma - VI

To find the area of the nanogon we integrate (or find the area under the curve) round the perimeter in sequence of the vertices. Let $(x_1, y_1) = (2,9); (x_2, y_2) = (6,11); (x_3, y_3) = (9,8);...(x_9, y_9) = (2,1)$ and $\color{#3D99F6}{(x_{10}, y_{10}) = (x_1, y_1) = (2,9)}$ . Then the area $A$ of the nanogon is given by:

$\begin{aligned} A & = \sum_{n=1}^9 \color{#3D99F6}{\frac{(x_{n+1}-x_n)(y_n+y_{n+1})}{2} \quad \quad \small \text{Area of a trapezium.}} \end{aligned}$

Using an Excel spreadsheet below, we can easily find that $A=\boxed{54.5}$ .