Bishop's Paradise
Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?
Problem
Which of the following statements is false?
A An octagon has $20$ diagonals
B A hexagon has $9$ diagonals
C A hexagon has $4$ more diagonals than a pentagon
D A pentagon has the same number of diagonals as it has sides
E A quadrilateral has twice as many diagonals as it has sides
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Drawing the diagonals for each of the shapes and counting shows that an octagon has $20$ diagonals, a hexagon has $9$, a pentagon has $5$ and a quadrilateral has $2$.
This can be used to show that A to D are all correct. A quadrilateral has half as many diagonals as it has sides, not twice as many, so statement E is false.
Alternatively, each vertex in a polygon shares a diagonal with $n-3$ others, if there are $n$ vertices, since it does not share one with itself or either of its neighbours. There are $n$ vertices, so this is $n(n-3)$. But this means we have counted each diagonal twice, so there are $\frac 12 n(n-3)$ in total. This gives the numbers obtained directly above.