1. goteachmaths.co.uk Interior & Exterior Angles of Polygons
– Complete Lesson
Check and delete unwanted slides.
Notes are at the bottom on some slides.

2. Which pairs of numbers go together to make 180?65 1 35 63
145 30 90
115 45
150 25
132
179 48 90
155
135
Which number is the odd one out?

3. Which pairs of numbers go together to make 180?65 1 35 63
145 30 90
115 45
150 25
132
179 48 90
155
135
Which number is the odd one out?

4. Interior & Exterior Angles of Polygons
08 November 2018
Interior & Exterior Angles of Polygons

5. Polygon Exterior Angle Hexagon Interior Angle Irregular Hexagon
Greek: Poly- "many“, -gon "angle"
Regular =
sides/angles all equal
Exterior Angle
Hexagon
Interior Angle
Irregular Hexagon
Pentagon
How can we calculate interior and exterior angles of polygons?

6. We know the interior angles of a triangle total 180°.
60°
How can we work out the
interior angles of a quadrilateral?
50°
70°
If we divide it into 2 triangles,
we can see the interior angles
must total 360°.

7. Worksheet

8. Sum of triangle’s interior angles.
Finding the Sum of Interior Angles in Different Polygons
Draw a line from the dot to each vertex (corner).
Count how many triangles are made from the shape.
Complete the table.
Name of shape
Number of sides (n)
Number of triangles
Sum of triangle’s interior angles.
Total
Triangle 3 1
1 x 180° =
180°
Quadrilateral
Any polygon
This can be printed full-page or displayed for students.
What is the rule to find the sum of interior angles of a polygon?
Extension: How large is each interior angle in these shapes?

9. Sum of triangle’s interior angles.
Finding the Sum of Interior Angles in Different Polygons
Draw a line from the dot to each vertex (corner).
Count how many triangles are made from the shape.
Complete the table.
Name of shape
Number of sides (n)
Number of triangles
Sum of triangle’s interior angles.
Total
Triangle 3 1
1 x 180° =
180°
Quadrilateral 4 2
2 x 180° =
360°
Pentagon 5
3 x 180° =
540°
Hexagon 6
4 x 180° =
720°
Any polygon
Half-completed worksheet to guide students.
What is the rule to find the sum of interior angles of a polygon?
Extension: How large is each interior angle in these shapes?

10. Sum of triangle’s interior angles.
Finding the Sum of Interior Angles in Different Polygons
Draw a line from the dot to each vertex (corner).
Count how many triangles are made from the shape.
Complete the table.
Name of shape
Number of sides (n)
Number of triangles
Sum of triangle’s interior angles.
Total
Triangle 3 1
1 x 180° =
180°
Quadrilateral 4 2
2 x 180° =
360°
Pentagon 5
3 x 180° =
540°
Hexagon 6
4 x 180° =
720°
Heptagon 7
5 x 180° =
900°
Octagon 8
6 x 180° =
1080°
Nonagon 9
7 x 180° =
1260°
Decagon 10
8 x 180° =
1440°
Any polygon n
n-2
(n-2) x 180
Completed investigation.
What is the rule to find the sum of interior angles of a polygon?
Extension: How large is each interior angle in these shapes?

11. Sum of Interior Angles = 180(n-2)
We can see that every shape can be divided into triangles.
The amount of triangles is two less than the number of sides (n).
An irregular octagon can be
divided into 6 triangles.
A pentagon can be divided into 3 triangles.
Total Interior Angles = 3 x 180° = 540°
Total Interior Angles = 6 x 180° = 1080°
Summary of findings
The formula for any regular or irregular polygon is:
Sum of Interior Angles = 180(n-2)

12. Sum of Interior Angles = 180(n-2)
We can see that every shape can be divided into triangles.
The amount of triangles is two less than the number of sides (n).
An irregular octagon can be
divided into 6 triangles.
A pentagon can be divided into 3 triangles.
Total Interior Angles = 3 x 180° = 540°
Total Interior Angles = 6 x 180° = 1080°
No animation.
The formula for any regular or irregular polygon is:
Sum of Interior Angles = 180(n-2)

13. 𝑥 𝑦 𝑦= ? 𝑥= ? 𝑧= ? 𝑧 𝑡=? 𝑡 125° 100° 80° 120° 100° 100° 95° 140° 160°
70°
150° 𝑡
80°

14. 𝑥 𝑦 𝑦=120° 𝑥=60° 𝑧=120° 𝑡=135° 𝑧 𝑡 125° 100° 80° 120° 100° 100° 95°
Total = 360°
Total = 540° 𝑥
125° 𝑦
100°
80°
120°
100°
𝑥=60°
𝑦=120°
100°
95°
𝑧=120°
Total = 720°
𝑡=135°
140°
160° 𝑧
70°
150° 𝑡
80°
1080°÷8

15. How does the ship need to turn to avoid land?
60°

16. 90° 90° 90° 90° What if the ship wants to go in a rectangle?
The ship must turn through 360°
90°
90°
90°
90°

17. 72° 72° 72° 72° 72° What if the ship wants to go in a pentagon?
It will be 360° in total, so how big is each turn?
360 ÷ 5 = 72°
72°
72°
72°
72°
72°

18. 60° 60° 60° 60° 60° 60° What if the ship wants to go in a hexagon?
It will be 360° in total, so how big is each turn?
360 ÷ 6 = 60°
60°
60°
60°
60°
60°
60°

19. What if the ship wants to go in an irregular route?
The turns will still total 360°.
20°
50°
100°
80°
40°
70°

20. Interior angle + exterior angle = 180°
Exterior angles
total 360°
Interior angles
Summary
Interior angle + exterior angle = 180°

21. Interior angle + exterior angle = 180°
Exterior angles
total 360°
Interior angles
No animation
Interior angle + exterior angle = 180°

22. We can make the pentagon smaller and smaller
and see what happens to the exterior angles.
Use your mouse wheel to zoom in and out.
So, exterior angles of a polygon sum to 360°

23. We can make the pentagon smaller and smaller
and see what happens to the exterior angles.
So, exterior angles of a pentagon sum to 360°

24. We can make the pentagon smaller and smaller
and see what happens to the exterior angles.
No animation.
So, exterior angles of a pentagon sum to 360°

25. 18° What is the size of this What is the size of this exterior angle?
This is one vertex of a regular polygon.
How many sides does this polygon have?
A pentadecagon has 15 sides.
What is the size of each exterior angle?
18°

26. 72° 45° 20 sides 18° 24° What is the size of this
exterior angle?
What is the size of this
exterior angle?
72°
45°
This is one vertex of a regular polygon.
How many sides does this polygon have?
A pentadecagon has 15 sides.
What is the size of each exterior angle?
20 sides
18°
24°

27. Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Handout to print.

28. Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Answers

29. Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
120°
60°
Quadrilateral 4
90°
Pentagon 5
72°
108°
Hexagon 6
Heptagon 7
51.4°
128.6°
Octagon 8
45°
135°
Nonagon 9
40°
140°
Decagon 10
36°
144°
Answers

30. Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
These are exterior angles of regular polygons.
How many sides does the polygon have?
These are exterior angles of regular polygons.
How many sides does the polygon have? a b c a b c
36°
36°
20°
20°
Worksheet to print
30°
30° d d e e
12°
12°
24°
24°
For each of the shapes:
What is the size of the interior angle?
What is the sum of interior angles?
For each of the shapes:
What is the size of the interior angle?
What is the sum of interior angles?

31. Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
120°
60°
Quadrilateral 4
90°
Pentagon 5
72°
108°
Hexagon 6
Heptagon 7
51.4°
128.6°
Octagon 8
45°
135°
Nonagon 9
40°
140°
Decagon 10
36°
144°
These are exterior angles of regular polygons.
How many sides does the polygon have?
10, 144°, 1440°
12, 150°, 1800°
18, 160°, 2880°
15, 156°, 2340°
30, 168°, 5040° a b c
36°
20°
Worksheet to print
30° d e
12°
24°
For each of the shapes:
What is the size of the interior angle?
What is the sum of interior angles?

32. 𝑥 𝑦 Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
These are the interior angles of regular polygons.
How many sides does the polygon have?
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10 c a b
120°
150°
160° d e
170°
175°
Find the sum of interior angles for each polygon.
These are exterior angles of regular polygons.
How many sides does the polygon have? a
What is the size of angle 𝑥? b c
36° 𝑥
20°
Worksheet to print
A decagon and a dodecagon (12 sides) with
equal length sides have been placed together.
What is the size of angle 𝑦?
30° d e
12° 𝑦
24°
For each of the shapes:
What is the size of the interior angle?
What is the sum of interior angles?

33. Complete the table for regular polygons.
Remember! Interior angle + exterior angle = 180°
These are the interior angles of regular polygons.
How many sides does the polygon have?
6, 720°
12, 1800°
18, 2880°
36, 6120°
72, 12600°
Shape
Sides
Exterior Angle
Interior Angle
Triangle 3
120°
60°
Quadrilateral 4
90°
Pentagon 5
72°
108°
Hexagon 6
Heptagon 7
51.4°
128.6°
Octagon 8
45°
135°
Nonagon 9
40°
140°
Decagon 10
36°
144° c a b
120°
150°
160° d e
170°
175°
Find the sum of interior angles for each polygon.
These are exterior angles of regular polygons.
How many sides does the polygon have?
10, 144°, 1440°
12, 150°, 1800°
18, 160°, 2880°
15, 156°, 2340°
30, 168°, 5040° a
What is the size of angle 𝑥? b c
36° 𝑥
20°
A decagon and a dodecagon (12 sides) with
equal length sides have been placed together.
What is the size of angle 𝑦?
30° d e
12° 𝑦
24°
𝑥 = 90°
𝑦 = 57°
For each of the shapes:
What is the size of the interior angle?
What is the sum of interior angles?

34. Find the exterior angles of these regular polygons. b b c c d d
Find the value of x.
Find the value of x. e f
70° e f
70°
40°
85°
40°
85°
80°
80° x° x°
65°
65°
100°
100°
75°
75° g g x° x°
35°
35°
30°
30°
85°
85°
70°
70°
What size is one exterior angle of
a regular icosagon?
(An icosagon has 20 sides)
What size is one exterior angle of
a regular icosagon?
(An icosagon has 20 sides)
75°
75° x° x°

35. Find the exterior angles of these regular polygons.b
72°
60°
45°
30°
110°
95°
65° c d
Find the value of x. e f
70°
40°
85°
80° x°
65°
100°
75° g x°
35°
30°
85°
70°
18°
What size is one exterior angle of
a regular icosagon?
(An icosagon has 20 sides)
75° x°

36. I can calculate the exterior angles of different regular polygons.
Check your success!
I can calculate the exterior angles of different regular polygons.
Given an exterior angle, I can calculate the number of sides of a regular polygon.
I can complete shape problems using rules for interior and exterior angles.