#mathmatrivia

So how big does the inner circle need to be in order to put those four points on one straight line?

It turns out to be about 38%. But this is math. We want to know exactly how much and we want to know how to find the answer!

Let's consider the bigger circle a unit circle, so R = 1. Then consider this triangle.

Ten equal sectors define the entire 360°, so they're 36° each. The center-touching angle of this triangle encompasses 2 of those sectors. Therefore, 72°.

Unfortunately, I have to use cosines now. Those who fear trigonometry may want to look away.

Cosine equals adjacent over hypotenuse. We defined the hypotenuse to be 1, because it's a unit circle. So the cosine of 72° is the height of that triangle divided by one. Or just the height of that triangle. h = cos 72°

Now lets look at this slightly different triangle.

That's only 1 out of 10 equal sectors that make up the entire circle. That's a 36° angle. And the cosine of that one is the height of the triangle (the same as the other triangle) divided by the radius of the inner circle — exactly we're trying to find! cos 36° = h / r But we know what h is. cos 36° = cos 72° / r Now we solve for r. r⋅ cos 36° = cos 72° r = cos 72° / cos 36°

That's the exact answer: cos 72° / cos 36°

If you throw it into a calculator, you'll get something like ≈ 0.3819660112510519363...

Like I said, about 38%. Or 38.197%, if you want to be pedantic. Which I hope you do.

This divide symbol ÷ is called an obelus. It was first used for mathematical division by Swiss Mathematician Johann Rahn in his 1659 book Teutsche Algebra (meaning "German Algebra"). Before that, it was a symbol used in the hand-copying of documents to indicate that some passage might need to be removed; the process was called obelism. The obelus is a common division sign in English, but can be unknown or mean different things in cultures that don't speak English.

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More information about:

Swiss Mathematician Johann Rahn https://mathshistory.st-andrews.ac.uk/Biographies/Rahn/

How to do square roots on paper.

I also made an animation with pause and play buttons here. Reload that page to see it calculate the square root of a different pseudorandom number.

A degenerate triangle occurs when all three vertices of a triangle lie on the same line. It is as if you took a plain old triangle and flattened it out.

When the sum of the two smaller side lengths is equal to the longest side length, it forms a degenerate triangle with angles 0°, 0°, and 180°.

But are they really triangles?

They have three sides. Check They have three angles. Check Do they really have three vertices? When one angle is 180°, does that really count as a vertex? They have an area of zero. That's better than undefined or indeterminate, but it's not a positive number. Does that count? The three angles sum to 180°. Check They fit on a plane. Check Triangles are supposed to be two dimensional shapes. Degenerate triangles only have one dimension.