# Unit 15 Section 2 : Angle Properties of Polygons

In this section we calculate the size of the interior and exterior angles for different regular polygons.
In a regular polygon the sides are all the same length and the interior angles are all the same size.

The following diagram shows a regular hexagon: Note that, for any point in a polygon, the interior angle and exterior angle are on a straight line and therefore add up to 180°.
This means that we can work out the interior angle from the exterior angle and vice versa:
 Interior Angle = 180° – Exterior Angle
 Exterior Angle = 180° – Interior Angle
If you follow around the perimeter of the polygon, turning at each exterior angle, you do a complete turn of 360°.
 In every polygon, the exterior angles always add up to 360°
Since the interior angles of a regular polygon are all the same size, the exterior angles must also be equal to one another.
To find the size of one exterior angle, we simply have to divide 360° by the number of sides in the polygon.
 In a regular polygon, the size of each exterior angle = 360° ÷ number of sides In this case, the size of the exterior angle of a regular hexagon is 60° because 360° ÷ 6 = 60° and the interior angle must be 120° because 180° – 60° = 120°
This also means that we can find the number of sides in a regular polygon if we know the exterior angle.
 In a regular polygon, the number of sides = 360° ÷ size of the exterior angle
We can use all the above facts to work out the answers to questions about the angles in regular polygons.

Example Question 1
A regular octagon has eight equal sides and eight equal angles.

(a) Calculate the size of each exterior angle in the regular octagon.
We do this by dividing 360° by the number of sides, which is 8.
The answer is 360° ÷ 8 = 45°.

(b) Calculate the size of each interior angle in the regular octagon.
We do this by subtracting the size of each exterior angle, which is 45°, from 180°.
The answer is 180° – 45° = 135°.

Example Question 2
A regular polygon has equal exterior angles of 72°.

(a) Calculate the size of each interior angle in the regular polygon.
We do this by subtracting the exterior angle of 72° from 180°.
The answer is 180° – 72° = 108°.

(b) Calculate the number of sides in the regular polygon.
We do this by dividing 360° by the size of one exterior angle, which is 72°.
The answer is 360° ÷ 72° = 5 sides.

Practice Question
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The interior angles of a regular polygon are all equal to 140°.

(a) What is the size of each of the exterior angles in the regular polygon? (b) How many sides does the polygon have? (c) What is the name of the polygon? ## Exercises

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Question 1
Calculate the size of the exterior angles of a regular polygon which has interior angles of:
Question 2
Calculate the sizes of the exterior and interior angles of:
 (a) a regular decagon exterior = °  interior = °  (b) a regular pentagon exterior = °  interior = °  Question 3
A dodecagon is a 12-sided polygon.

Calculate the size of the exterior angle of a regular dodecagon.
°  Question 4
The exterior angle of a particular regular polygon is 12°.

How many sides does this polygon have?  Question 5
Calculate the number of sides of a regular polygon with interior angles of:
Question 6
Each exterior angle of a regular polygon is 4°.

(a) How many sides does the polygon have?
sides  (b) What is the size of each interior angle in the polygon?
°  (c) What is the total of all the interior angles added together?
°  You have now completed Unit 15 Section 2
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