Explainer: Interior Angles of a Polygon

In this explainer, we will learn how to find the sum of the interior angles of a polygon given the number of its sides and the measure of an angle in a regular polygon.

We will use this skill to then calculate the interior angles of a regular polygon. Both of these techniques will be shown in context by solving various geometry problems.

Definition: Polygons

A polygon is a simple 2-dimensional closed shape made up of straight line segments.

The number of sides and the number of angles in a polygon are equal, and this number is usually used to classify the shape.

You may be familiar with the names of simple polygons. As an example, a three-sided polygon is called a triangle. The table below shows the general name for an 𝑛-sided polygon for 2<𝑛≀10.

Number of SidesName
3Triangle
4Quadrilateral
5Pentagon
6Hexagon
7Heptagon (septagon)
8Octagon
9Nonagon
10Decagon

You may note that a polygon with less than three sides cannot be formed, since this shape would require curved line segments!

To begin, we will learn how to calculate the sum of the interior angles, π‘†οŠ, of a polygon with 𝑛 sides.

Let us start with the simplest polygon, which is a triangle (𝑛=3). You should be familiar with the fact that the sum of the interior angles in a triangle is always 180∘.

In order to prove this, we can imagine a general triangle, with interior angles angles π‘ŽοŠ§, π‘ŽοŠ¨, and π‘ŽοŠ©. By extending each side beyond the vertices, we can form the exterior angles π‘οŠ§, π‘οŠ¨, and π‘οŠ©.

An interesting property worth noting is that the external angles of a polygon will always sum to 360∘.

This can be illustrated by considering an arrow at point 𝑃, pointing in the direction parallel to the base of the triangle. Let us imagine this arrow traveling clockwise around the perimeter of the triangle, turning through each of the external angles (π‘οŠ§, π‘οŠ¨, and π‘οŠ©). Upon arriving back at point 𝑃, the arrow will have completed one complete turn of 360∘: 𝑏+𝑏+𝑏=360.∘

Now let us observe that each interior angle and its corresponding exterior angle share a vertex and form one side of a straight line. Given this information, we know each pair will add up to 180∘: π‘Ž+𝑏=180,π‘Ž+𝑏=180,π‘Ž+𝑏=180.∘∘∘

We can therefore form an equation for the sum of the internal and external angles: (π‘Ž+𝑏)+(π‘Ž+𝑏)+(π‘Ž+𝑏)=3(180)=540.∘∘

We can now rearrange this to form an equation for the sum of the interior angles in triangle, π‘†οŠ©, by substituting in the sum of the exterior angles found earlier: (π‘Ž+π‘Ž+π‘Ž)+(𝑏+𝑏+𝑏)=540(𝑆)+(360)=540𝑆=180.∘∘∘

Based on this method, we can develop a general proof for the sum of the interior angles of an 𝑛-sided polygon.

Definition: Sum of Interior Angles of a Polygon

Consider an 𝑛-sided polygon with 𝑛 vertices. The diagram below shows an example for 𝑛=6.

Each vertex will have an associated interior and exterior angle. Since each angle pair share a vertex and form one side of a straight line, they will sum to 180∘: π‘Ž+𝑏=180,π‘Ž+𝑏=180,…,π‘Ž+𝑏=180.∘∘∘

This allows us to form an equation for the sum of the interior and exterior angles for an 𝑛-sided polygon: (π‘Ž+𝑏)+(π‘Ž+𝑏)+β‹―+(π‘Ž+𝑏)=(𝑛)180.

In order to find a relationship between the sum of the interior angles and 𝑛, we can first recall that the exterior angles of a polygon will always sum to 360∘: 𝑏+𝑏+β‹―+𝑏=360.∘

This allows us to substitute the sum of the external angles into the previous equation and simplify: (π‘Ž+π‘Ž+β‹―+π‘Ž)+(𝑏+𝑏+β‹―+𝑏)=(𝑛)180,(π‘Ž+π‘Ž+β‹―+π‘Ž)+(360)=(𝑛)180(π‘Ž+π‘Ž+β‹―+π‘Ž)+2(180)=(𝑛)180(π‘Ž+π‘Ž+β‹―+π‘Ž)=(π‘›βˆ’2)180.

Finally, we define the sum of the interior angles (π‘Ž+π‘Ž+β‹―+π‘Ž) as π‘†οŠ to reach a formula: 𝑆=(π‘›βˆ’2)180.

This formula allows us to find the sum of the interior angles of an 𝑛-sided polygon.

Let us now look at a few examples of finding the interior angles of a polygon using the formula we have found.

Example 1: Finding the Sum of the Interior Angles of a Hexagon

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

Answer

In order to find the sum of the internal angles of an 𝑛-sided polygon, π‘†οŠ, we can use the following formula: 𝑆=(π‘›βˆ’2)180.

A hexagon is a a polygon with six sides and six vertices.

In this case, we define 𝑛=6. We can hence substitute this value into the equation and simplify to find the sum of the interior angles: 𝑆=(6βˆ’2)180=(4)180=720.∘

The sum of the internal angles of a hexagon is 720∘.

Example 2: Finding the Measure of Angle of a Pentagon given the Measures of the Other Angles

In the figure, if π‘šβˆ π΄=137∘, π‘šβˆ π΅=78∘, π‘šβˆ πΆ=113∘, and π‘šβˆ πΈ=131∘, find π‘šβˆ π·.

Answer

The diagram above shows a pentagon with four of the five angles given. We know that the sum of the interior angles for this shape, π‘†οŠ«, can be expressed as follows: π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ πΆ+π‘šβˆ π·+π‘šβˆ πΈ=𝑆.

We can substitute in the given angle values into the equation: 137+78+113+π‘šβˆ π·+131=𝑆459+π‘šβˆ π·=𝑆.

In order to find the sum of the internal angles in a pentagon, we can use the general formula for the sum of the interior angles of an 𝑛-sided polygon, π‘†οŠ: 𝑆=(π‘›βˆ’2)180.

A pentagon is a polygon with five sides and five vertices and we can therefore substitute 𝑛=5 into the equation: 𝑆=(5βˆ’2)180=(3)180=540.∘

We can now substitute our value for π‘†οŠ« into our original equation for shape 𝐴𝐡𝐢𝐷𝐸: 459+π‘šβˆ π·=540.

Finally, we solve for π‘šβˆ π·: π‘šβˆ π·=540βˆ’459=81.∘

Example 3: Finding the Measure of an Angle in a Quadrilateral given the Other Angles’ Measures

In the figure, segments 𝐸𝐷 and 𝐡𝐺 meet at 𝐴. If π‘šβˆ πΉ=90∘, π‘šβˆ πΊ=97∘, π‘šβˆ πΈ=97∘, π‘šβˆ π·=137∘, and π‘šβˆ πΆ=95∘, find π‘šβˆ π΅.

Answer

Let us first observe that the shape 𝐴𝐸𝐹𝐺 is a quadrilateral with three known angles and one unknown angle (𝐸𝐴𝐺).

We can therefore form an equation for the sum of the internal angles in quadrilateral 𝐴𝐸𝐹𝐺 and substitute in the known values: π‘šβˆ πΈπ΄πΊ+π‘šβˆ πΊ+π‘šβˆ πΉ+π‘šβˆ πΈ=π‘†π‘šβˆ πΈπ΄πΊ+97+90+97=π‘†π‘šβˆ πΈπ΄πΊ+284=𝑆.οŠͺοŠͺοŠͺ

You may already be familiar with the fact that the sum of the internal angles in a quadrilateral is equal to 360∘. This can also be proven using the formula for the sum of interior angles in a polygon: 𝑆=(π‘›βˆ’2)180.

Substituting for 𝑛=4, we reach the familiar result 𝑆=(4βˆ’2)180=(2)180=360.οŠͺ∘

We can now input our value for 𝑆οŠͺ into the original equation for the internal angles in quadrilateral 𝐴𝐸𝐹𝐺 and solve for the unknown angle 𝐸𝐴𝐺: π‘šβˆ πΈπ΄πΊ+284=360π‘šβˆ πΈπ΄πΊ=360βˆ’284=76.∘

We can now observe that angle 𝐸𝐴𝐺 and angle 𝐷𝐴𝐡 are vertically opposite angles formed by the intersection of the straight lines 𝐸𝐷 and 𝐡𝐺.

Given this information, we know the measures of the two angles will be equal, and we can state the following: π‘šβˆ π·π΄π΅=π‘šβˆ πΈπ΄πΊ=76.∘

We have now set up the same situation for quadrilateral 𝐴𝐡𝐢𝐷, where three angles are known and one angle (𝐡) is unknown. Following the same steps, we form an equation for the angles in 𝐴𝐡𝐢𝐷: π‘šβˆ π·π΄π΅+π‘šβˆ π΅+π‘šβˆ πΆ+π‘šβˆ π·=𝑆.οŠͺ

We can then use the fact that 𝑆=360οŠͺ∘ and substitute in the known values: 76+π‘šβˆ π΅+95+137=360.

Finally, we solve to find π‘šβˆ π΅ by simplifying our equation: π‘šβˆ π΅=360βˆ’76βˆ’95βˆ’137=52.∘

Now that we know how to find the sum of the interior angles of a polygon, let us see how this technique can be applied to regular polygons to find the individual angle values.

Definition: Finding the Interior Angle of a Regular Polygon

A polygon is considered to be regular when all its angles are equal and all its sides are equal. In any other case, the polygon is considered to be irregular.

The diagram below shows a regular hexagon in comparison to an irregular hexagon (𝑛=6).

For a regular polygon, we can find the measure of one interior angle 𝐴 by diving the sum of the interior angles π‘†οŠ by the number of angles: 𝐴=𝑆𝑛.

Note that this is only true for a regular polygon, since all 𝑛 angles are equal.

We have shown that the sum of the interior angles for any 𝑛-sided polygon can be found using the following formula: 𝑆=(π‘›βˆ’2)180.

We can therefore substitute for π‘†οŠ to find the formula for the measure of the internal angles in a regular polygon (in terms of 𝑛): 𝐴=(π‘›βˆ’2)180𝑛.

It is worth noting that the above relationship implies that the value of 𝐴 is necessarily less than 180∘ for a regular polygon. Any solutions greater than this value should be double-checked, and you should ensure the shape in question is indeed a regular polygon.

This fact can be seen by imagining a regular arrangement of angles greater than 180∘ that form a closed shape. In such a case, the angles would lie outside of the shape and hence would not be classified as the β€œinterior angles”!

For an irregular polygon, interior angles may be greater than 180∘, but this it outside the scope of this explainer.

As a final note, two line segments that form an angle of 180∘ cannot be considered two sides of a polygon since they cannot be distinguished from a single segment. For this reason 180∘ itself is not counted as angle in this situation.

Let us look at some questions which use this formula for the interior angles of a regular polygon.

Example 4: Finding the Number of Sides of a Polygon given the Measures of Its Interior Angles

Each interior angle of a polygon is 144∘. How many sides does it have?

Answer

For this question, we have been given the measure of each interior angle of a regular polygon (𝐴) and are asked to find the number of sides and, therefore, the number of angles (𝑛).

In order to find (𝑛), we can substitute the known value 𝐴=144 into the formula for the interior angle in a regular polygon: 𝐴=(π‘›βˆ’2)180𝑛144=(π‘›βˆ’2)180𝑛.

We can now solve this equation by first multiplying both sides by 𝑛 and then grouping and simplifying the terms: 144𝑛=(π‘›βˆ’2)180144𝑛=180π‘›βˆ’360βˆ’36𝑛=βˆ’360𝑛=10.

We have now found the number of sides (and angles) in this shape is 10. We therefore have a regular decagon.

You may be asked to consider the interior angles of regular polygons (𝐴) in conjunction with the sum of the angles in a polygon (𝑆).

Example 5: Finding the Measure of an Angle Using Supplementary Angles

Given that 𝐴𝐡𝐢𝐷𝐸 is a regular pentagon, find π‘šβˆ π΄π΅π‘‹.

Answer

Observing the diagram, we can see that π‘šβˆ π΄π΅πΆ, π‘šβˆ π΄π΅π‘‹, and π‘šβˆ π‘‹π΅π‘Œ together are the angles around one side of the straight line πΆπ΅π‘Œ.

The sum of the angles that share a vertex around one side of a straight line is 180∘. This allows us to define the following relationship: π‘šβˆ π΄π΅πΆ+π‘šβˆ π΄π΅π‘‹+π‘šβˆ π‘‹π΅π‘Œ=180.∘

In order to solve this question, we will first work to determine π‘šβˆ π΄π΅πΆ and π‘šβˆ π‘‹π΅π‘Œ, using the fact that both of these are interior angles of a polygon. Following this, we will be able to take the above angle relationship and solve for π‘šβˆ π΄π΅π‘‹.

Let us first find π‘šβˆ π΄π΅πΆ by using the formula for the interior angles (𝐴) of the regular polygon. To do so, we can use the following formula: 𝐴=(π‘›βˆ’2)180𝑛.

We can find the interior angles in a regular pentagon by setting 𝑛=5. In other words, π‘šβˆ π΄π΅πΆ will be equal to 𝐴, since it forms one of the five angles in the regular pentagon 𝐴𝐡𝐢𝐷𝐸: π‘šβˆ π΄π΅πΆ=𝐴=(5βˆ’2)1805=(3)1805=5405=108.∘

We can now work on finding π‘šβˆ π‘‹π΅π‘Œ using the fact that we know the sum of the interior angles in a triangle; namely, 𝑆=180∘. Using this information, we can set up the following equation for triangle π΅π‘‹π‘Œ: π‘šβˆ π‘‹π΅π‘Œ+π‘šβˆ π΅π‘‹π‘Œ+π‘šβˆ π‘‹π‘Œπ΅=180.

Since we have been given two of the three angles in triangle π΅π‘‹π‘Œ, we can substitute in these values and solve for the unknown angle: π‘šβˆ π‘‹π΅π‘Œ+79+64=180π‘šβˆ π‘‹π΅π‘Œ=180βˆ’79βˆ’64=37.∘

Finally, we can substitute the two values we have found, π‘šβˆ π΄π΅πΆ=108∘ and π‘šβˆ π‘‹π΅π‘Œ=47∘, into the original equation for angles around one side of a straight line. This will allow us to solve for π‘šβˆ π΄π΅π‘‹: 108+π‘šβˆ π΄π΅π‘‹+37=180π‘šβˆ π΄π΅π‘‹=180βˆ’108βˆ’37=35.∘

We have now answered the question by finding that π‘šβˆ π΄π΅π‘‹=35∘.

Key Points

  1. A polygon is a simple 2-dimensional closed shape made up of straight line segments.
  2. The sum of the external angles of a polygon will always be 360∘.
  3. The sum of the interior angles, π‘†οŠ, of an 𝑛-sided polygon can be found using the following formula: 𝑆=(π‘›βˆ’2)180.
  4. A polygon is considered to be regular when all its angles are equal and all its sides are equal. In any other case, the polygon is considered to be irregular.
  5. The internal angles of a regular polygon will always be less than 180∘, whereas the interior angles of an irregular polygon may be greater than 180∘.
  6. The measure of the internal angles, 𝐴, of a regular 𝑛-sided polygon can be found using the following formula: 𝐴=(π‘›βˆ’2)180𝑛.

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