Angle between two rectangles rotated around a point with a gap inbetween

I am trying to find the angle between two rectangles when there is a known gap between them.

See this diagram:

I have simplified the problem into three triangles, two of which are the same.

Here is my attempted solution:

$$\theta = 2\alpha + \beta$$ Finding $\alpha$, this is the angle of a right angle so I can just use $tan$ \begin{align} \tan\alpha &= \frac{a}{2r}\\ \therefore \alpha &= \arctan{\frac{a}{2r}} \end{align}

That's easy, when I test $\alpha$ seems reasonable.

Finding $\beta$, this isn't a right angle triangle so I use the cosine rule as I know all three sides:

\begin{align} b^2 &= 2r^2 + 2r^2\cos\beta\\ b^2 - 2r^2 &= 2r^2\cos\beta\\ \frac{b^2 - 2r^2}{2r^2} &= \cos\beta\\ \frac{b^2}{2r^2} - 1 &= cos\beta\\ \therefore\beta &= \arccos\left(\frac{b^2}{2r^2} - 1\right) \end{align}

When I now check my $\beta$ value with some reasonable numbers I get an unreasonable angle?

For example: \begin{aligned} \text{let } b &= 4\\ r &= 128\\ \\ \therefore\beta &= \arccos\left(\frac{4^2}{2\times128^2} - 1\right)\\ &\approx 0.99\pi \end{aligned}

Why am I getting such a big angle, I expect it to be more like $0.01\pi$!

Update

The comments highlighted a mistake in my diagram, the lengths around $\beta$ aren't of length $r$.

So my new formula for $\beta$ is:

\begin{align} r_2^2 &= \left(\frac{a}{2}\right)^2 + r^2\\ \therefore r_2 &= \sqrt{\left(\frac{a}{2}\right)^2 + r^2}\\ \\ \beta &= \arccos\left(\frac{b^2}{2\left(\frac{a}{2}^2 + r^2\right)} - 1\right)\\ &= \arccos\left(\frac{2b^2}{w^2+4d^2} - 1\right) \end{align}

But I still get a ridiculous result!

• Your sketch is wrong: The hypotenuse in the triangle with the angle $\alpha$ cannot have length $r$ since one of the cathesuses has length $r$. So you need to introduce a different length, say $r_{2}$ for the hypothenuse. – Sebastian Dec 4 '13 at 10:50
• It's either that or the triangle does not have a right-angle. – Sebastian Dec 4 '13 at 10:54
• @Sebastian Ahh of course. The triangles around $\alpha$ are definitely right angles though. I can calculate $r_2$ with Pythagoras so it doesn't make things too much more complicated. – Lerp Dec 4 '13 at 11:48
• Just from the sketch it is not clear whether the length of the side of the triangle which is part of the rectangle outline is $\frac{a}{2}$ - if that is the case, you should include this in the problem statement. – Sebastian Dec 6 '13 at 12:43

Assuming that the line originating at the point $X$ bisects the line of lenght $a$ of the rectangle in a right angle and that we know $r$, $a$ and $b$. We can calculate $r_{2}$ (like you did) using Pythagoras' Theorem. Now consider the triangle with the sides $b$, $r_{2}$ and $r_{2}$ in your sketch. Then the line which meets $b$ in a right angle and goes through $X$ bisects $b$ (it is just the height line because our triangle is equal-sided). So \begin{equation*} \sin{\frac{\beta}{2}} = \frac{\frac{b}{2}}{r_{2}}\end{equation*} and hence \begin{equation*} \beta = 2 \cdot \arcsin{\frac{b}{2 \cdot r_{2}}} = 2 \cdot \arcsin{\frac{b}{2 \cdot \sqrt{r^{2} + \left(\frac{a}{2}\right)^{2}}}}\end{equation*}