Angles in Polygons

You have probably heard many times that Angles in a triangle add up to 180 degrees and that angles in a quadrilateral add up to 360 degrees. If you haven’t, this is your reminder that angles in a triangle add to 180 degrees and angles in a quadrilateral add to 360 degrees. However, have you ever wondered what angles in a five, six or even seven-sided shape sum to? What if we had a 24 sided shape? Okay, you have probably not. Regardless, in this article, we will be exploring angles in polygons. However, we must first outline what we mean by ‘polygon’.

Angles in Polygons Angles in Polygons

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Table of contents

    The term ‘poly’ means many, so a polygon is just a shape with many sides. When we say ‘many’, we mean three or more. So essentially, a polygon can be any 2D shape that's not a circle. A polygon is a regular polygon if all sides and Angles are the same.

    Interior Angles in Polygons

    When we talk about what angles add up to a polygon, we refer to the sum of interior angles. We will use this term a lot from now on, so it is essential to know it.

    Angles in Polygon, Polygon with Interior angles Labelled, Jordan MadgeAngles in Polygons- Polygon with interior angles labelled, Jordan Madge- StudySmarter Originals

    For a polygon, an interior angle is an angle inside the polygon (see the diagram above). The sum of interior angles is what all of the angles inside the polygon add up to. So, formally, we already know that the sum of interior angles in a triangle is 180° and in a quadrilateral is 360°.

    Sum of Interior Angles Formula

    Previously, we have just been expected to know that the interior angles in a triangle sum to 180° and the interior angles in a quadrilateral sum to 360°. We have just taken it as a fact and have never really questioned it. However, you may now be thinking, why is this the case? Or you may not... However, a convenient formula tells us the sum of interior angles for any polygon. It goes as follows...

    For any given polygon with n sides,

    Sum of Interior Angles= (n-2)×180°

    So, when we have a triangle, n=3 and so the sum of interior angles is (3-2) × 180= 180°.

    Similarly, when we have a quadrilateral, n=4 and so the sum of interior angles is (4-2)×180=360°

    We already knew those two results. However, now we can apply this formula to shapes with more than four sides.

    Calculate the sum of interior angles for a pentagon.

    Solution:

    A pentagon has five sides, so using the formula, the sum of interior angles is (5-2)×180=540°

    Calculate the sum of interior angles for a nonagon.

    Solution:

    A nonagon has nine sides, so using the formula, the sum of interior angles is (9-2)×180=1260°

    Calculate the sum of interior angles for the shape below.

    Angles in Polygons, Example with 14 sided polygon, Jordan MadgeAngles in Polygons- 14 sided polygon, Jordan Madge- StudySmarter Originals

    Solution:

    The above shape has 14 sides and so the sum of interior angles is (14-2)×180=2160°

    Calculate the sum of interior angles for a 24 sided shape.

    Solution:

    When id="2555885" role="math" n=24, the sum of interior angles is (24-2)×180=3960°

    Calculate the size of the angle x in the image below.

    Angles in Polygons, Example with quadrilateral, Jordan MadgeAngles in Polygons- quadrilateral example, Jordan Madge- StudySmarter Originals

    Solution:

    This shape has five sides, so the sum of interior angles is (5-2)×180=540°

    Each of the right angles in the shape is 90°and so we can work out the missing angle by subtracting all of the given angles from 540. Thus, x= 540-90-90-90-130=140°

    Table of Common Interior Angles

    The below table shows the sum of interior angles for the first eight polygons. However, you could confirm these results for yourself using the formula.

    Shape # sides Sum of Interior Angles (°)
    Triangle 3180
    Quadrilateral 4360
    Pentagon 5540
    Hexagon 6720
    Heptagon 7900
    Octagon 81080
    Nonagon 91260
    Decagon 101440

    Calculating Each Interior Angle

    Earlier, we defined regular polygons as polygons with equal sides and angles. We therefore may wish to calculate each interior angle of a regular polygon. We first calculate the sum of interior angles and divide this Number by the Number of sides.

    Calculate each interior angle for a regular hexagon.

    Solution:

    Using table 1, we can see that the sum of interior angles for a hexagon is 720°. Since this hexagon is regular, each of the angles is the same and thus we can work out each interior angle by dividing 720 by 6. Therefore, each interior angle is 120°.

    Below is part of a tiling pattern consisting of three regular pentagons. Calculate the angle labelled x.

    Angles in Polygon, Pentagon Example, Jordan MadgeAngles in Polygons- Pentagon Example, Jordan Madge- StudySmarter Originals

    Solution:

    The sum of interior angles for each regular hexagon is 720° (using the table of common interior angles).

    Thus, each interior angle in each hexagon is 120°.

    Angles in Polygon, Pentagon Example, Jordan MadgeAngles in Polygons- Pentagon Example, Jordan Madge- StudySmarter Originals

    Recall that angles around a point sum to 360 degrees. Therefore, x can be found by subtracting the other known angles from 360. Thus, x=360-108-108=144°

    Exterior Angles in Polygons

    There is also an exterior angle for each interior angle in a polygon. An exterior angle is formed between any side of the shape and the straight line extended outside of the shape. This may sound not very clear, but it is easier to see illustrated.

    Angles in Polygon, Polygon with Interior and Exterior angles Labelled, Jordan MadgeAngles in Polygons- Pentagon with interior and exterior angles labelled, Jordan Madge- StudySmarter Originals

    In the diagram above, the interior angles are labelled orange, and the exterior angles are green. Since the exterior angle lies on the same straight line as the interior angle, the sum of the interior and exterior angles is 180°. Therefore, an exterior angle can be calculated by subtracting the interior angle from 180°.

    In the image below, angles x and y are exterior angles. Calculate x and y.

    Angles in Polygons, Example with exterior angles, Jordan MadgeAngles in Polygons- Pentagon with interior and exterior angles, Jordan Madge- StudySmarter Originals

    Solution:

    For exterior angle x, the interior angle is 109°. Thus, since angles on a straight line add up to 180°, x=180-109=71°. Angle y is another exterior angle and since angles on a straight line add to 180, y=180-81=99°.

    Calculate each exterior angle of a regular heptagon.Solution: A heptagon has seven sides and thus the sum of interior angles is 900°Since this heptagon is regular, we can work out each interior angle by dividing 900 by 7 to obtain 128.6°. Therefore, we can calculate each exterior angle by subtracting this from 180. Thus, each exterior angle is 180-128.6=51.4°.

    A heptagon is also sometimes referred to as a septagon.

    Sum of Exterior Angles

    The sum of exterior angles for any polygon is dead simple. It is 360°. Unlike interior angles, we do not need to memorise any fancy formulae to work out the sum of exterior angles; we simply need to remember the sum of exterior angles for any polygon 360°. Using this, we can begin to answer some more questions.

    Each exterior angle of a regular polygon is 10. Work out the number of sides the polygon has.

    Solution:

    Since the sum of exterior angles is 360°, and each exterior angle is 10°, we can calculate the number of sides by 360÷10=36. Thus, this polygon has 36 sides.

    Each interior angle of a regular polygon is 165. Work out the number of sides the polygon has.

    Solution:

    If each interior angle is 165, each exterior angle must be 180-165=15°. Since the sum of the exterior angles is 360°, there must be 360÷15=24 sides.

    Angles in polygons - Key takeaways

    • The interior angles in a polygon are the angles inside the polygon.
    • To calculate the sum of interior angles, subtract two from the number of sides and multiply the result by 180 degrees.
    • If the polygon is regular, each of the sides is the same.
    • An exterior angle is formed between any side of the shape and the straight line extended outside of the shape.
    • The sum of exterior angles of any polygon is 360 degrees, regardless of the number of sides.
    Frequently Asked Questions about Angles in Polygons

    What do angles in a polygon add up to?

    It is different for each polygon. The sum of interior angles in a regular polygon can be found by subtracting two from the number of sides and then multiplying this result by 180 degrees. 

    What is the sum of the exterior angles of a polygon?

    The sum of exterior angles is 360 degrees for any polygon. 

    What is the formula for the sum of the interior angles of a polygon?

    (n-2) x 180

    What is the sum of the interior angles of a polygon?

    The sum of interior angles in a regular polygon can be found by subtracting two from the number of sides and then multiplying this result by 180 degrees. 

    How to find the missing angle in a polygon?

    First work out what the sum of angles should be, and then subtract the angles that you know to work out the missing one. 

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