If the measures of two interior
angles of a polygon are 120 degrees and 40 degrees and the sum of the rest of the
angle measures is 380 degrees, find the number of sides.
We’ve been given the measures of
two of the interior angles of a polygon and the sum of the measures of the remaining
angles. Using this information, we need to
work out how many sides the polygon has.
To do this, we need to recall that
there is a general formula for the sum of the interior angles in any polygon. The sum of the interior angle
measures in an 𝑛-sided polygon, which we denote as 𝑆 sub 𝑛, is equal to 𝑛 minus
two multiplied by 180 degrees. This is derived from the fact that
an 𝑛-sided polygon can be divided into 𝑛 minus two triangles. And as the sum of the interior
angles in a triangle is 180 degrees, the sum of the interior angles in the polygon
is 𝑛 minus two multiplied by 180 degrees.
We can calculate the sum of the
interior angles in this polygon by summing the two individual angle measures and the
sum of the remaining angle measures to give 540 degrees. We can then form an equation by
equating this value with the expression in the formula we already wrote down. And as both sides are measured in
degrees, we don’t need to include the units. We have 𝑛 minus two multiplied by
180 equals 540. To solve this equation for 𝑛, we
begin by dividing both sides by 180 to give 𝑛 minus two is equal to three. Finally, we can add two to each
side of the equation to give 𝑛 equals five.
Using the general formula for the
sum of the interior angle measures in an 𝑛-sided polygon, we’ve found that this
polygon has five sides.