Diagonals
How many diagonals does a regular icosagon (20 sides) have?
Age
14 to 16
Challenge level
Problem
A square has 2 diagonals, a regular pentagon has 5 and a regular hexagon has 9.
How many diagonals does a regular icosagon (20 sides) have?
This problem is adapted from the South East Asian Mathematics Competition
Student Solutions
Answer: 170
Counting the diagonals drawn from each corner
Image
20 corners means 17 diagonals from each corner, total 17$\times$20 diagonals
But then each diagonal is counted twice, once at each corner
So there are 17$\times$20$\div$2 = 170 diagonals.
Counting by drawing the diagonals
Starting at a corner, it will be joined to every other corner by a diagonal, with the exception of the two adjacent corners, we will draw $17$ diagonals from this corner.
The next corner around will also be joined to every other corner but the two adjacent corners (one of which is the first corner), so we will draw $17$ diagonals from this corner.
The next corner will also be joined to every other corner but the two adjacent corners (one of which is the second corner), but it is already joined to the first corner, so we will draw $16$ diagonals from this corner.
The next corner will also be joined to every other corner but the two adjacent corners (one of which is the third corner), but it is already joined to the first and second corners, so we will draw $15$ diagonals from this corner.
This pattern will continue until we have drawn on all of the diagonals - so there will be a total of $17+17+16+15+14 +...+ 1$ diagonals.
$17+17+16+15+14 +...+ 1=17+(17+16+15+14 +...+ 1)$, and $$\begin{align}&17+16+15 +...+ \hspace{1mm}3\hspace{1mm}+\hspace{1mm}2\hspace{1mm}+\hspace{1mm}1\\+&\hspace{2mm}1\hspace{1mm}+\hspace{1mm}2\hspace{1mm}+\hspace{1mm}3\hspace{1mm} +...+ 15+16+17\end{align}\\$$$$
\begin{align}&=\underbrace{18+18+18 +...+ 18+18+18}_\text{17 times}\\
&=18\times17\end{align}$$
So $17+17+16+15+14 +...+ 1=17+\frac{1}{2}(18\times17)=170$.