Crooked Hourglass

It's entirely possible that I missed some shortcuts and made this too complicated, but here's how I solved it:

The area of whole square is $r^2$ , which can also be represented as $a+b+2c$ , so $a+b+2c=r^2$ .

$a + c = \dfrac{\pi r^2}4 \Rightarrow c = \dfrac{\pi r^2}4 - a$

Combining these gives

$a + b + 2 \left( \dfrac{2\pi r^2}4 - a\right) = r^2 \Rightarrow b = a + r^2 - \dfrac{\pi r^2}2 = a + r^2 \left( 1 - \dfrac\pi2 \right) .$

This gives us the relationship between $a$ and $b$ . This relationship will be the same for any value of $r$ . If we knew a value of $a$ or $b$ that corresponded with a specific value of $r$ , we could use these to solve for the remaining variable, making this a much shorter problem. I am not aware of a way to do this, hence the second half of my solution:

Finding the area of region $b$ seemed easiest to me. It is simply the area of the square, minus two "pie slices", minus the area of a triangle - all things I can solve for somewhat easily. Starting with the triangle seems best. It is an equilateral triangle, because each of its sides is simply the radius of the circles, $r$ . (Equilateral triangles have angles of 60 degrees. Keep this in mind, as it will be useful later). Using the Pythagorean theorem :

$\left( \dfrac r2\right)^2 + h^2 = r^2 \Rightarrow h = \dfrac{r \sqrt3}2 .$

Area of a triangle:

$\text{Area} = \dfrac12 \times \text{ base } \times \text{ height } = \dfrac12 r \left( \dfrac{r\sqrt3}2 \right) = \dfrac{\sqrt3}4 r^2 .$

Next we find the area of a "pie slice" - which is a certain fraction of a circle. Because each angle of the triangle is 60 degrees, the angle of the circle-piece is 30 degrees. Dividing 30 degrees by 360 degrees gives us the "fraction" of a circle we have: $\dfrac{1}{12}$ .

Two of these is $\dfrac{1}{6}$ of a circle of radius $r$ , so their combined area is:

$\dfrac{1}{6}\pi r^2$

The area of region $b$ :

$b=r^2-\dfrac{1}{6}\pi r^2-\dfrac{\sqrt{3}}{4}r^2$

From here, we simply chose a value for $r$ . I chose 2 in order to get rid of some fractions. Putting 2 into the above equation for $b$ , we find:

$b=4-\dfrac{2π}{3}-\sqrt{3}$

Plugging this and our value $r = 2$ into the equation for $a$ , we find

$a=\dfrac{4}{3}\pi-\sqrt{3}$

At this point, it's just a matter of using a calculator to divide $a$ by $b$ . You should get that $\dfrac{a}{b}\approx 14.155$ .