Today we will continue our discussion from last week about polygons and cover interior angles. (You may want to review AngleExterior before continuing). This concept appeared 4 times last year, with a median placement at question # 49.
Number Dojo Level: 211
We have already established that:
- The sum of the exterior angles of any polygon is 360°, and
- At any vertex, the exterior angle and the interior angle are supplementary (their measures add up to 180°).
How to solve:
To find the measure of one interior angle, we can first find the exterior angle and subtract it from 180°. Or we can use the formula 180(n - 2)/n, where n is the number of sides of the polygon.
To find the sum of the measures of the interior angles of a polygon, we can use the formula 180(n - 2) because we are multiplying 180(n - 2)/n by n.
Example 1: The measure of an interior angle of a regular hexagon is ___°
- A hexagon has 6 sides, so its exterior angle measures 360/6 or 60°.
- The interior angle is (180 - 60) = 120°.
Example 2: The measure of an interior angle of a regular decagon is ___°
- A decagon has 10 sides, so its exterior angle measures 360/10 or 36°.
- The interior angle is (180 - 36) = 144°.
Example 3: The interior angle of a regular icosagon measures ___°
- An icosagon has 20 sides, so its exterior angle measures 360/20 or 18°.
- The interior angles is (180 - 18) = 162°.
Example 4: The sum of the interior angles of a regular octagon is ___°
- An octagon has 8 sides.
- 180(n - 2) = 180(8 - 2) = 180(6) = 1080°.
Example 5: The sum of the measures of the interior angles of a 20-sided polygon is ___°
- 180(20 - 2) = 180(18) = 3240°.
Example 6: The number of sides of a regular polygon with an interior angle measure of 156° is ___
- If the interior angle is 156°, the exterior angle is (180 - 156) = 24°.
- n = 360/24 = 15.