 Interior and exterior angles of polygons

Lesson

Sum of interior angles of a polygon

We already know that the sum of angles in a triangle is $180$180°.

What about the sum of angles in a quadrilateral? Well a quadrilateral can be thought of as two triangles, so the internal angle sum is $180+180=360$180+180=360°

This interactive will show you different interior angle sums for different polygons.  You can also see how many triangles fit into it.

Watch this video if you would like to see this interactive in action -

If $n$n represents the number of sides, then the number of triangles in any polygon is $n-2$n2.

If each triangle has $180$180°, then the total interior angle sum of a polygon is $180\times\left(n-2\right)$180×(n2)

Sum of Exterior Angles of a Polygon

Why is the sum of the exterior angles of any convex polygon always 360°?

Why is the result the same for a 3-sided polygon (i.e. a triangle) as for a 20-sided polygon?

To see why, imagine you are driving a car around this polygonal track.  What is the total of the angles you turn when doing 1 complete lap.  These angles are the external (exterior) angles of the polygon.  What is the sum?

Make sure you keep the polygon convex (no concave structures). See here to remind yourself the difference between convex and concave.

Watch this video if you would like to see this interactive in action -

Worked Examples

Question 1

Consider the pentagon shown

a) What is the interior angle sum of a pentagon?

b) Find an expression for the interior angle sum of the pentagon involving the exterior angles $a$a, $b$b, $c$c, $d$d, and $e$e

c) Hence solve for the sum of the exterior angles of the pentagon.  That is solve for the value of $a+b+c+d+e$a+b+c+d+e

d) Is the sum of exterior angles of any polygon equal to $360^\circ$360°

Question 2

Find the value of $y$y in the given diagram.

Question 3

Consider the non-convex hexagon in the figure shown.

a) What is the least number of triangles that it can be divided into?

b) What is the interior angle sum of the non-convex hexagon?

Outcomes

9D.MG3.01

Determine, through investigation using a variety of tools and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons

9D.MG3.02

Determine, through investigation using a variety of tools and describe some properties of polygons, and apply the results in problem solving