GCSE Maths: How To Calculate Angles In Polygons
Introduction
At GCSE Foundation and Higher Levels you are required to be able to solve questions that involve angles in polygons. The level of knowledge is similar but obviously the difficulty of the questions on a higher paper will be a little more challenging.
A polygon is a many sided shape. They are two dimensional and are made from straight lines. You can have a regular polygon which means that all the lengths and angles are the same. There are also irregular polygons which means that the length and angles are not equal.
What is important is being able to successfully calculate the sum of interior angles of a polygon and the process is the same whether you have a regular polygon or irregular polygon.
There are some polygons which are more than likely to be familiar with depending on the number of sides that it has:
Number of sides | Name of polygon |
3 | Triangle |
4 | Quadrilateral |
5 | Pentagon |
6 | Hexagon |
8 | Octagon |
Each polygon has exterior angles and regardless of the number of sides, the sum of exterior angles is always 360°. This is an important fact to remember.
You can, for a regular polygon, work out the size of each exterior angle by using the formula E=\frac{360}{n}, where 360° is the sum of exterior angles and n is the number of sides and E is the exterior angle.
You can also determine the sum of the interior angles of a regular polygon by using the formula 180(n-2), where n is the number of sides.
The sum of an interior and exterior angle adds up to 180°.
Angles In Polygons - An Example
Take a look at the following question:
In order to answer this question you need to remember the formula E=\frac{360}{n}.
\text { Here } E=30^{\circ} \therefore 30=\frac{360}{n} \rightarrow 30 n=360 \therefore n=12So this is the number of sides that the polygon has.
Another Angles In Polygons Example
Take a look at the following angles in polygons question:
To find the size of an exterior angle you need to know the size of each interior angle.
There are 9 sides and 9 interior angles. But what does the sum of all the interior angles equal to? You can answer this by using the formula 180 \times(n-2). There are 9 sides so the sum of interior angles is 180 \times 7-1260^{\circ}.
Because there are 9 angles of equal size (dealing with a regular polygon) this means that the calculation to find the size of each interior angle is 1260 ÷9=140°.
The sum of an exterior and interior angle is 180° so each exterior angle is 180°-140°=40°.
You can check if this is correct because the sum of all exterior angles is 360°. 40° ×9=360°.
These are so far quite straightforward questions and these can appear on either the GCSE Foundation or the GCSE Higher Maths Papers. Should you feel that you need some additional support with regards to Polygons or indeed any other maths related topic at GCSE then you can use the services of an experienced online maths tutor. They are qualified teachers and have extensive experience of teaching and deep understanding of the curriculum.
Question Practice
Try the following angles in polygons question on your own before looking at the solution.
Question Practice Solution
So how did you get on? Hopefully you got the correct answer of 84°.
This is a slightly more challenging question than what we have seen so far and would more than likely appear on a Higher GCSE Paper.
In this question the first thing that you will see is that no numbers are presented to you and this alone can be off putting.
You need to use the diagram and the information discussed to help you answer this question.
There are two regular pentagons (5 sided shapes) and one of these will have a sum of interior angles of 180(n-2)=180(5-2)=540^{\circ}
So the interior angle of a pentagon is, E=\frac{540}{5}=108^{\circ}
The triangle is equilateral so each interior is 60°.
The sum of these angles is 276°.
But this is still not the answer for the angle DEF.
The final part of the calculation is to be able to recognise that the angle at E is that at a point i.e. 360° ∴ 360°-276°=84°
It is hoped that the solution makes sense. Try the question again on your own to see if you understand the techniques that we have mentioned. During our intense GCSE Maths Revision Courses that take place during the half term holidays it is these types of questions that are looked at as well as other challenging angles in polygons questions from other areas of maths.
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