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Polygons

1. Naming Polygons

A  '3 sided' polygon is called a 'triangle'
A  '4 sided' polygon is called a 'quadrilateral'
A  '5 sided' polygon is called a 'pentagon'
A  '6 sided' polygon is called a 'hexagon'
A  '7 sided' polygon is called a 'heptagon'
A  '9 sided' polygon is called a 'nonagon'
A '10 sided' polygon is called a 'decagon'

The number of sides is usually referred to as 'n'
i.e. n = 5 means a 'pentagon'

2. Regular and Irregular Polygons

Some of the rules below apply to ALL polygons (regular or irregular). Other rules only apply to regular polygons...

3. All Polygons (Regular & Irregular):

e.g. For a Triangle:

Sum of the interior angles = 180(3-2) = 180°

e.g. For a Quadrilateral:

Sum of the interior angles = 180(4-2) = 360°

e.g. For a Pentagon:

Sum of the interior angles = 180(5-2) = 540°

e.g. Find the missing exterior angle in this polygon

First of all, we have to write in all of the interior angles

The angle at the left (100°) is an exterior angle. The interior angle corresponding to this is '180°-100°' = 80°

Similarly, the interior angle on the right is 120°. So our polygon looks like this:

Now, since the number of sides ('n') = 5:

			Sum of Interior Angles = 180(5-2) = 540
 
			So:  80° + 110° + 120° + 90° + x  = 540°
 
					        400° + x  = 540°
 
			=> 			       x  = 140°
 
 

e.g. Find 'x' and hence label all of the interior angles in this polygon

Again, we start by finding all of the interior angles:

Now, since the number of sides ('n') = 6:

			Sum of Interior Angles = 180(6-2) = 720°
 
	 3x° + (180-x)° + (180-2x)° + 4x° + (180-x)° + 3x = 720°
 
			=>			540° + 3x = 720°
 
			=>				x = 30°
 

So the interior angles are (starting from the top left and working anticlockwise): 90°, 90°, 150°, 120°, 120°, 150°.

e.g. A polygons interior angles add up to 2520°. How many sides does it have?

Since:

			Sum of Interior Angles = 180(n-2)
 
		   =>			  2520 = 180(n-2)
 
		   =>			    16 = n
 
 

Note: Rules 1 & 2 will work for any polygon; regular or irregular

e.g. Find the missing angle in this polygon

This time, we need to find all of the 'exterior' angles. The exterior angle corresponding to the 110° angle at the top is '180°-110°' = 70° etc...

And, since the sum of the exterior angles must equal 360°:

			Sum of Exterior Angles = 360°
 
	     70° + 100° + (180-x)° + 90° + 60° = 360°
 
	 =>			       500 - x = 360
 
	 => 				     x = 140°
 
 

e.g. Find 'x' and hence label all of the exterior angles in this polygon

Again, we need to find all of the exterior angles:

And since the sum of the exterior angles is 360°:

					   Sum of Exterior Angles = 360°
  
		(180-3x)° + (180-3x)° + x° + (180-4x)° + 2x° + x° = 360
 
	   =>						 540 - 6x = 360
 
	   => 							x = 30°
 

So the exterior angles are (starting from the top left and working anticlockwise): 90°, 90°, 30°, 60°, 60°, 30°

4. Regular Polygons:

Note: These rules will work ONLY work for a regular polygon

e.g. Sketch a heptagon and label all of the angles (angle at centre, interior angle and exterior angle

The angle at the centre is: 360/7 = 51.4°
So the exterior angle is: 51.4°
Which means the interior angle is: 180 - 51.4 = 128.6°

e.g. Find the angles labelled 'x' and 'y' in this regular polygon:

Since this is a regular octagon; we know the angle at the centre must be 360/8 = 45°

So therefore, the exterior angle must also be 45°

And the interior angle must be 180° - 45° = 135°:

So, we can say immediately that x = 135° (because it is an interior angle)

Looking at triangle OED (which is isosceles). ÐEOD = 45° (angle at centre). So ÐOED = 67.5°

Looking at the trapezium ABCD (which is an isosceles trapezium): ÐABC = ÐBCD = 135°. And ÐCDA = ÐBAD = y°

But these 4 angles must add up to 360 (angles in a quadrilateral add up to 360°):

			Sum of Interior Angles = 180(4-2) = 360°
  
				    135° + 135° + y° + y° = 360
 
			   =>		         270 + 2y = 360
 
			   => 			        y = 45°
 
 

e.g. A regular polygon has interior angle = 165.6°. How many sides does it have?

So, exterior angle = angle at centre = 180° - 165.6° = 14.4°

  And since:	Angle at Centre = 360°
				   n
 
  			   14.4 = 360°
				   n
 
 		       =>     n = 25
 
 

e.g. A regular pentagon has sides of 4cm. What is the area of the pentagon. Could an irregular polygon with the same perimeter have a greater area?

We can find the area of the pentagon by first dividing it into 5 isosceles triangles
Extracting one of these, we can split in half to get a right angled triangle:

We can find the side opposite the 54° angle using trigonometry:

			  ╖
		A = 2 cm  ║		O = A tan θ
		O = ?	  ║		O = 2 tan 54°
		θ = 54°   ║		O = 2.753 cm
			  ╜
 

And then use Area = ½ × base × height to find the area of the right angled triangle:

				Area = ½ × base × height
 
				Area = ½ ×  2   ×  2.753
 
				Area =      2.753 cm²
 

Multiply this by '2' to get the area of the isosceles triangle:

				Area of isosceles triangle = 5.51 cm²
 

And multiply this by '5' to get the area of the pentagon:

				Area of Pentagon = 27.5 cm²
 

A circle is the shape that gives the largest area for its perimeter and for polygons, a regular polygon is the closest to a circle. So the answer is 'NO' an irregular polygon with the same perimeter could not have a greater area

5. To Draw a Regular Polygon (e.g. a pentagon)

The best way to draw accurately a regular polygon is to draw it within a circle:

1. Start by drawing a circle in pencil

2. Draw a line from the centre of the circle to the circumference (called a 'radius') in pencil

3. Find the 'Angle at the centre' 
   Angle at Centre = 360°/n      [in this case: 360°/5 = 72°]

4. Draw another radius at 72° (in this case) to the first one

5. Repeat until you've gone all the way around the circle

6. Using a pen, connect together these points on circumference with straight lines to form your pentagon

7. Rub out the pencil lines which helped you to draw the pentagon

Question 1: Using RULE 2: Sum of exterior angles = 360°

				Sum of exterior angles = 360°
 
			     90° + 85° + 65° + p + 90° = 360°
 
			                             p = ...
 
 

Question 2: Again, we are going to use the RULE 2: Sum of exterior angles = 360°

But one of the angles (115°) is an interior angle

If we extend the horizontal line to the left, then the exterior angle can be found as '180 - 115' = 65°

				Sum of exterior angles = 360°
  
			  65° + ...° + ...° + p + ...° = 360°
 
						    p  = ...°
 
 

Question 3: Still using RULE 2:

				Sum of exterior angles = 360°
  
				    ...° + 2x + x + 2x = 360°
 
						     x = ...
 
 

Question 6: We know from RULE 4 that:

				Exterior Angle = Angle at Centre
 
					  72°  = Angle at Centre
 

And, we know from RULE 3 that:

				Angle at Centre  =  360 
						     n 
 
					     72° =  360 
						     n 
 

Solving...

Alternative Method: Imagine there are 'n' sides to this polygon. Since the sum of the interior angles must equal 360°:
72n = 360°
So n = ...

Question 7: Using RULE 1, for a polygon with 5 sides:

			Sum of Interior Angles = 180(5-2) = 540°
  
				     x + 2x + 3x + 3x + x = 540°
 
							x = ...
 
 

Question 12: We want to use RULE 1 (for a polygon with 4 sides):

			Sum of Interior Angles = 180(4-2) = 360°
  

BUT, one of these angles (‘x’) is an exterior angle...

We can find the interior angle at that point using RULE 5:

			Interior Angle = 180 - Exterior Angle
 
			Interior Angle = 180 -        x
 

Now, using RULE 1 as described above:

			Sum of Interior Angles = 180(4-2) = 360°
  
			       ...° + ...° + ...° + 180-x = 360°
 
							x = ...
 
 

Question 13: Again, we need to find the two interior angles, where we have been given exterior angles...

And then use RULE 1 (for a polygon with 5 sides):

			Sum of Interior Angles = 180(5-2) = 540°