How would you determine the length of this line segment tangent to a circle?

Question

So I am working on rebuilding my hot tub siding and want to convert the rounded corners to flat surfaces, but need to determine the width across. Of course the easy answer is measure it, but I wanted to figure it out mathematically and was having trouble figuring it out. It would look something like:

I'm looking to find the length of the red line. My current measurements put the radius at 6.5 inches. How would I go about doing this?

Answer 1

This is the side of regular $Octagon$:

Answer 2

$2(\sqrt{2}-1)R$

Hint: the diagonal of the square is $\sqrt{2}R$.

Answer 3

Hint

$\tan(\frac {\pi} 4)= \frac x {2h}$

Where $h=R(\sqrt 2 -1)$

Answer 4

You can do a bit of algebraic reasoning to rework the diagram into what you see here:

Notice the burnt orange dashed lines and their corresponding lengths. Essentially, we’ve broken the red segment into the hypotenuses of two right triangles. Using the definitions of the trigonometric functions, we get

$$\sin\theta = \frac{r-r\cos\theta}{A}$$ $$\sin\phi = \frac{r-r\sin\phi}{B}$$

There is a plethora of different forms in which you can write this using trigonometric identities, but I will leave that to you.

You said $r = 6.5 \ \mathrm{in}$ and I assume that $\theta=\phi=\frac\pi2=45^\circ$, but this isn’t necessarily true. It is true, however, that $\theta$ and $\phi$ are complementary.