I'm not going to be the one checking those calculations! 😎👍

Correct, the area of the magenta region equals $10$.

If we look at the upper blue region:

![image](https://i.ibb.co/MpJPs5g/one.png)

The area is the sum of the components:

$$
A = g * (t + \frac{s}{2}) + u*s + u*t
$$

Adding the same (with $$h$$ instead of $$g$$) for the lower blue region:

$$
38 = (g+h)*(t + \frac{s}{2}) + 2*u*s + 2*u*t
$$

The apothem formula gives

$$
tan(\frac{\pi}{24})*(g+h+2*u) = s
$$

Simple trig says

$$
t = s*cos(15°)
$$
$$
u = s*sin(15°)
$$

Those last four equations solve for $$s \approx 1.82625$$, or total area of polygon $$A=152$$.

As seen in @0xbitcoiner's drawing, the total area is the sum

$$
38 + 47 + 47 + 2 * x = 152
$$

Therefore, $$x = 10$$
