GCSE Maths: Polygons And Angles
The “poly” means many so the word “polygon” refers to a many-sided shape.
It is possible to determine the size of polygons and angles using two simple techniques. One is to use the fact that the interior of triangle is 180° and to also use a formula.
Polygons and Angles: The Sum Of Interior Angles Using Triangles
Suppose we have the following 4 sided shape:
What we want to be able to do is to work out the sum of the interior angles. Well what we can do is actually split the shape so we have two triangles as shown below:
Because there are now two triangles and given that the sum of the interior angle of a triangle is 180° then the sum of the quadrilateral (4 sided shape) is 180° x 2 = 360°
Polygons and Angles - What About A Pentagon?
Just as before with the triangle what we can do is split the shame into triangles and this is shown by the black lines. From the diagram you will see that there are three triangles each with a sum of interior angles of 18 degrees. So the total sum of interior angles for the pentagon is 180 x 3 = 540.
Finding the sum of interior angles for any sided polygon
It will not always be easy to draw/sketch a shape especially if the number of sides is very large.
Thankfully there is a formula that allows us to work out the sum of the interior angles for a shape of any side.
The formula is: 180° ×(n-2) where n is the number of sides.
Let us see if this formula works for a triangle, quadrilateral and pentagon.
Triangle
The number of sides is 3, so 180 x ( 3 – 2) = 180 x 1 = 180°
Quadrilateral
The number of sides is 4, so 180 x (4 – 2) = 180 x 2 = 360°
Pentagon
The number of sides is 5, so 180 x (5 – 2) = 180 x 3 = 540°
So we can see that the formula is giving the correct rules.
If you should forget what the formula is then you can always try to find how many triangles there are in a shape. The only downside to this is that you could have a shape with many sides and this is not going to be practical to draw.
But having said this, the formula should not be too difficult to recall. Polygons and angles can and do feature a lot in maths exams and during any GCSE Maths Revision course you will see topics such as this and how it can be combined with areas of geometry.
Finding missing angles – Examples
Example 1: Find the missing angle “a”
Here we have a 4 sided shape so we know that the sum of interior angles can be worked out as follows:
180 x (4 – 2) = 180 x 2 = 360°
So this means that;
110 + 54 + 50 + a = 360
214 + a = 360
A = 146°
Example 2: Find the missing angle “d”
Here we have a 5 sided shape and so the sum of interior angles can be found as follows:
180 x ( 5 – 2 ) = 180 x 3 = 540°
And so the missing angle can be found as follows:
76 + 79 + 210 + 39 + d = 540
404 + d = 540
d = 136°
Overall Polygons can be quite fun to work out and as you have seen there are some questions that can involve the use of algebra. If you are looking to achieve a high grade in GCSE Maths then being highly skilled in algebra is something that you will need to be.
If algebra or any other topic is considered one of your strengths then you may consider the use of a GCSE online maths tutor who can help you improve your subject knowledge as well as your confidence.
To help with your continued GCSE maths revision here are some questions that you can try and you will find the answers at the bottom.
Multiple Choice Questions
A four sided shape has the angles; 135, 75, 60 and “a”. Find “a”.
75°
80°
66°
72°
A four sided shape has the angles; 80, 90, 90 and “b”. Find “b”.
115°
95°
110°
105°
A four sided shape has angles 122, 157, 80 and “f”. Find “f”.
55°
10°
180°
1°
A six sided shape has angles 110, 120, 95, 105, “b” and 120. Find “b”.
170°
165°
201°
55°
A six sided shape has angles 79, 65, 270, 69, 111 and “f”. Find “f”.
62°
126°
74°
185°
An eight sided shape has angles “h”, 150, 145, 128, 125, 135, 140, 143. Find “h”.
41°
411°
114°
35°
A seven sided shape has angle “k”, 125, 250, 96, 56, 87, 19. Find “k”.
267°
76°
456°
148°
A four sided shape has angles 2x, x + 30, x + 15, x – 10. What is the value of x?
77°
45°
21°
65°
A four sided shapes has angle y + 5, 2y, y – 35, y + 70. What is the value of y?
46°
64°
22°
87°
A four sided shape has angle 2y + 10, y + 65, 2y – 15, y + 25. What is the value of y?
46°
4523°
4623°
4523°
Answers:
1 – b; 2 – c; 3 – d; 4 – a; 5 – b; 6 – c; 7 – a; 8 – d; 9 – b; 10 – c
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