Interior and Exterior Angles Worksheets | Questions and Revision | MME

Interior and Exterior Angles Worksheets, Questions and Revision

Level 4-5

Interior and Exterior Angles

The interior angles of a shape are the angles inside the shape.

The exterior angles are the angles formed between a side-length and an extension. 

Rule: Interior and exterior angles add up to 180\degree.

Having the ability to rearrange equations will help with interior and exterior angle questions. 

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Exterior Angles

Rule: The exterior angle = \dfrac{360\degree}{\textcolor{red}{n}}

where \textcolor{red}{n} is the number of sides.

The sum of all the exterior angles will equal 360\degree.

how to calculate exterior angles

For the triangle shown, we can see it has \textcolor{red}{3} sides, so to calculate an exterior angle we do: 

\dfrac{360\degree}{\textcolor{red}{3}} = 120\degree

 

how to calculate exterior angles
Level 4-5

Interior Angles

Rule: Sum of interior angles = (\textcolor{red}{n} - 2) \times 180\degree

Where \textcolor{red}{n} is the number of sides. 

 

 

how to calculate interior angles

To find the sum of the interior angles for the triangle shown we do the following:

(\textcolor{red}{3} - 2) \times 180\degree = 180\degree

This means that

\textcolor{limegreen}{a} + \textcolor{limegreen}{b} + \textcolor{limegreen}{c} = 180\degree

Note: You can find the interior angle of a regular polygon by dividing the sum of the angles by the number of angles. You can also find the exterior angle first then minus from 180\degree to get the interior angle. 

how to calculate interior angles
Level 4-5

Example: Finding Interior and Exterior Angles

ABCD is a quadrilateral.

Find the missing angle marked x

[2 marks]

unknown interior angle example

 

This is a 4-sided shape, to work out the interior angles we calculate the following:

(\textcolor{red}{n}-2)\times 180 =360\degree.

Next we can work out the size of \angle CDB as angles on a straight line add up to 180\degree.

180 - 121 = 59\degree

Now we know the other 3 interior angles, we get that

x = 360 - 84 - 100 - 59 = 117\degree

unknown interior angle example
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Example Questions

This shape has 5 sides, so its interior angles add up to,

 

180 \times (5 - 2) = 540\degree

 

Hence each interior angle is,  

 

x\degree=540\degree \div 5 = 108\degree

This shape has 8 sides, so its interior angles add up to,

 

180 \times (8 - 2) = 1080\degree

 

Hence each interior angle is,  

 

x\degree=1080\degree \div 8 = 135\degree

This shape has 5 sides, so its interior angles must add up to

 

180 \times (5 - 2) = 540\degree.

 

We can’t find this solution with one calculation as we did previously, but we can express the statement “the interior angles add up to 540” as an equation. This looks like

 

33 + 140 + 2x + x + (x + 75) = 540

 

Now, this is a linear equation we can solve. Collecting like terms on the left-hand side, we get

 

4x + 248 = 540.

 

Subtract 248 from both sides to get

 

4x = 292.

 

Finally, divide by 4 to get the answer:

 

x = 292 \div 4 = 73\degree

This shape has 4 sides, so its interior angles add up to

180 \times (4 - 2) = 360\degree.

We don’t have any way of expression two of the interior angles at the moment, but we do have their associated exterior angles, and we know that interior plus exterior equals 180. So, we get

\text{interior angle CDB } = 180 - (y + 48) = 132 - y

Furthermore, we get

\text{interior angle CAB } = 180 - 68 = 112

 

Now we have figures/expressions for each interior angle, so we write the sum of them equal to 360 in equation form:

 

112 + 90 + 2y + (132 - y) = 360

 

Collecting like terms on the left-hand side, we get

 

y + 334 = 360

 

Then, if we subtract 334 from both sides we get the answer to be

 

y = 360 - 334 = 26\degree.

Worksheets and Exam Questions

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Drill Questions

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Interior angles of polygons