# Right Angled Octagon

Weekly Problem 38 - 2008

A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

## Problem

A quadrilateral can have four right angles. What is the largest number of right angles an octagon (8 sides) can have?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

$6$ right angles can be achieved, as in the diagram below:

Image

If there were seven right angles, these would be a total of $7 \times 90^\circ = 630^\circ$. The total interior angle of an octagon is $6 \times 180^\circ = 1080^\circ$, so the final angle would have to be $1080^\circ - 630^\circ = 450^\circ$. The interior angle cannot be more than $360^\circ$, so this cannot be achieved.