Video Transcript
If a regular polygon has an exterior angle of 90 degrees, find the number of sides it has.
We’re told that this polygon is regular. This means that all its sides are of equal length, and of greater importance here, all its angles are of equal measure. This is true for both interior and exterior angles. They’re not necessarily equal to each other, but all the interior angles have the same measure and all the exterior angles have the same measure. We can also recall that because the exterior angles of any polygon sum to 360 degrees, the measure of the exterior angle in a regular 𝑛-sided polygon is found by dividing 360 degrees by the number of sides.
We’re told that the exterior angle of this regular polygon is 90 degrees. And so we can use this knowledge to form an equation: 360 degrees over 𝑛 equals 90 degrees. As both sides are measured in degrees, we don’t need to include the units. We now solve this equation for 𝑛. The first step is to multiply both sides of the equation by 𝑛. This gives 360 equals 90𝑛. Next, we divide both sides of the equation by 90 to give 𝑛 equals four. We’ve therefore found that this regular polygon has four sides and is in fact a square.
This makes sense because we know that each interior angle in a square is 90 degrees. And each exterior angle, which is found by subtracting the measure of the interior angle from 180 degrees, is also 90 degrees. In fact, the square is the only regular polygon in which the interior and exterior angles are of equal measure. We’ve found that the regular polygon that has an exterior angle of 90 degrees has four sides, and hence it is a square.
