Finding the area of a fraction of a circle defined by points on a grid.

Question

A grid is shown below, where the shortest distance between any two points is 1. Find the area of the circle that passes through points $X,$ $Y,$ and $Z.$

Can it be assumed that the arc created over each of the chord lengths shown will create a semicircle when put together? How can I determine what fraction of the circle passes through points X,Y, and Z to calculate the area?

Answer 1

Let $X(3,3)$, $(0,0)$ and $Z(4,-4)$. Then, the area of the triangle XYZ is

$$A= \frac12| X_x(Y_y-Z_y) + Y_x(Z_y-X_y) + Z_x(X_y-Y_y)|= 12$$

The area is also given by

$$A= \frac{XY\cdot YZ\cdot ZX }{4R}$$

where $R$ is the circumradius. Thus, the area of the circumcircle is

$$Area= \pi R^2 = \pi\left( \frac{XY\cdot YZ\cdot ZX }{4A} \right)^2=\frac{\pi\cdot 18\cdot 32\cdot 50}{16\cdot 144}=\frac{25\pi}2 $$

Alternatively, recognize that XYZ is a right triangle with the hypotenuse $XZ= 5\sqrt2=2R$, which also leads to the area $\frac{25\pi}2$. But, the general approach above is applicable for arbitrary grid points.

Answer 2

As we know that a unique circle passes through three given points. Hence, if we assume the points $X, Y, Z$ as $(3, 3), (0, 0), (4, -4) $ the unique circle will be like this

Note: $X, Z$ are diametrically opposite points. Now, you can get answer.

Answer 3

Note that the distance $XY$ is $3$ units, where a unit is the length of the diagonal formed by the smallest square in the lattice, and $YZ$ is $4$ units. Does that strike anything?