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# need an explanation

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A square has two diagonals, and a convex pentagon has five diagonals. How many diagonals does a convex decagon have?

Jun 14, 2020

#1
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First, a decagon is a 10-gon. That means it has 10 sides, and 10 vertices.

We can think of it in a combinatorial way.

There are a total of $$C^{10}_2 = 45$$ ways to connect two different vertices. But connecting neighbouring vertices doesn't count, so we have to exclude the sides from our consideration. There are 10 sides, so we need to take away 10 ways from the 45 ways possible to get the number of diagonals.

Therefore, we get

$$\text{Number of diagonals in a decagon} = 35$$.

Generally, the number of diagonals in an n-gon is $$\dfrac{n(n - 3)}2$$. I will leave this as an exercise for you to prove.

Jun 14, 2020
#2
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thank you!!!

amazingxin777  Jun 14, 2020