Today we will get back to geometry and learn some more about polygons. (In the past couple of months, we have covered Squares, Rhombus, Rectangle, Trapezoid, and TriangleRight). Today we will discuss exterior angles. This concept appeared 12 times last year, with a median placement at question # 48.
Number Dojo Level: 202
First, let's define interior angles. An interior angle is an angle inside a polygon which is formed by two sides that share an endpoint. The easiest way to envision this is to think of a square. Each interior angle of a square is a right angle; it measures exactly 90°.
An exterior angle of a polygon is the angle formed by one side and extensing out its adjacent side. Envision a triangle standing on the ground. The exterior angle is the angle (outside the triangle) formed by one of its upright legs and the ground. The exterior angle is supplemental to the interior angle, so the two angles together form a straight line and add up to 180°.
For more discussion on both types of angles, see Wikipedia.
Types of Polygons:
As a review, you will need to know (memorize) how many sides are in each type of polygon:
Quadrilateral/Square/Rectangle/Rhombus/Trapezoid: 4 sides
Pentagon: 5 sides
Hexagon: 6 sides
Heptagon/Septagon: 7 sides
Octagon: 8 sides
Nonagon: 9 sides
Decagon: 10 sides
Undecagon: 11 sides
Dodecagon: 12 sides
Icosagon: 20 sides
First, we need to understand that the sum of the exterior angles of ANY polygon is 360°. To find the measure of a single exterior angle of a polygon, simply divide 360° by the number of sides of the polygon. For number sense purposes, assume that the polygon is regular (meaning every angle has equal measure), even if not stated specifically in the question.
There will be a few different variations of this concept; let's look at some examples.
Example 1: The sum of the exterior angles in a pentagon is ___ degrees
- The answer is 360, regardless of the shape of the polygon.
Example 2: A regular octagon has an exterior angle of measure ___ degrees
- An octagon has 8 sides. 360 ÷ 8 = 45.
Example 3: What is the measure of an exterior angle in a regular decagon? ___°
- A decagon has 10 sides. 360 ÷ 10 = 36.
Example 4: If a regular polygon has a side length of 4 and exterior angle of 40°, then its perimeter is ___
- Divide 360 by 40 to find that the polygon has 9 sides.
- Multiply 9 (the number of sides) by 4 (the side length) to get 36. This is the perimeter.
Example 5: The measure of an exterior angle of a regular heptagon is ___°
- A heptagon has 7 sides. 360 ÷ 7 = 360/7 or 51 3/7.