# Question Video: Finding the Measure of the Interior Angle of a Polygon Mathematics • 8th Grade

If the measures of two interior angles of a polygon are 120° and 40° and the sum of the rest of the angle measures is 380°, find the number of sides.

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### Video Transcript

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.