16. The curve $x^{2}+y^{2}=25$ is drawn. Points on the curve whose $x$-coordinate and $y$-coordinate are both integers are marked with crosses. All of those crosses are joined in turn to create a convex polygon $P$. What is the area of $P$ ?
16. 074 The curve $x^{2}+y^{2}=25$ is a circle of radius 5 centred at the origin.
The polygon $P$ has vertices at coordinates $(0,5),(3,4)$, $(4,3),(5,0),(4,-3),(3,-4),(0,-5),(-3,-4),(-4,-3)$, $(-5,0),(-4,3),(-3,4)$.
We may find the area of that part of $P$ in the upper-right quadrant by splitting it into two trapezia and a triangle as shown in the diagram. This area is
$$
\left(\frac{1}{2}(5+4) \times 3\right)+\left(\frac{1}{2}(4+3) \times 1\right)+\left(\frac{1}{2} \times 3 \times 1\right)=\frac{37}{2} \text {. }
$$

Therefore the area of $P$ is $4 \times \frac{37}{2}=74$.