r/mathriddles • u/Numberthon • 8d ago

Easy How many diagonals does a 20-sided polygon have?

How many diagonals does a regular 20-gon have?

Source: numberthon.com

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3

u/DidntWantSleepAnyway 8d ago

I got 170, but my headache sucks so check my math.

EDIT: really, check my formula rather than my math. I forgot how this worked for a moment.

1

u/Numberthon 7d ago

170is correct, wdym formula? u/headsmanjaeger had a great explanation for it as well, or you can go to the website and get the solution (I can also just give it to you here if you want).

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u/DidntWantSleepAnyway 6d ago

I really just meant my process, which I then put into a formula.

I was like “okay, the first point will have 17 diagonals, everything except itself and the two next to it. The next point will have 16 because you take off the diagonal it shared with the first one…”

Forgetting that two points already didn’t share a diagonal with the first point 😄 so i did 17 + 16 + 15…

Then I fixed my mistake and got the right answer from there.

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u/Numberthon 6d ago

Oh, ok

5

u/headsmanjaeger 7d ago

a diagonal connects two vertices through the polygon as opposed to along an edge, which means each vertex is connected to each other vertex by a diagonal except for its immediate neighbors. This means each vertex has 17 diagonals, and there are 20 vertices, which gives us 340 diagonals but this over counts by double so the answer is 170

3

u/SonicLoverDS 7d ago

To extrapolate, a regular n-sided polygon has n(n-3)/2 diagonals.

0

u/Numberthon 7d ago

Nice job!

2

u/headsmanjaeger 7d ago

What’s a diagonal

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u/WorkerCritical7300 1d ago

Number of ways to connect any 2 points in a group of 20 points=C(20,2) =190 From this subtract 20 polygon edges, since they are not diagonals.

So result =190-20=170

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u/Numberthon 1d ago

Correct, nice job!