Lesson Explainer: Interior Angles of a Polygon | Nagwa Lesson Explainer: Interior Angles of a Polygon | Nagwa

Lesson Explainer: Interior Angles of a Polygon Mathematics • First Year of Preparatory School

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

Definition: Polygons

A polygon is a simple two-dimensional closed shape made up of straight line segments called sides. Each point where two sides of a polygon meet is called a vertex (the plural is β€œvertices”).

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

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

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

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

Before we can work through some examples, we need a few more definitions to help us describe and classify different polygons.

Definition: Interior and Exterior Angles

An interior angle is an angle inside a polygon at one of its vertices. An exterior angle is an angle outside the polygon; it is formed by a side and the extension of an adjacent side.

At any given vertex, the measures of the interior and exterior angles sum to 180∘.

Definition: Convex and Concave Polygons

A convex polygon is a polygon with all interior angles measuring less than 180∘.

A straight line drawn through a convex polygon will intersect its sides exactly twice.

A concave polygon is a polygon with one or more interior angles measuring more than 180∘.

A straight line drawn through a concave polygon will intersect its sides more than twice.

Let us start with a simple example.

Example 1: Identifying a Polygon as Concave or Convex

Is this polygon concave or convex?

Answer

Recall that a convex polygon is a polygon with all interior angles measuring less than 180∘, while a concave polygon is a polygon with one or more interior angles measuring more than 180∘.

From the diagram, we see that the interior angles of this polygon have measures that range from 90∘ up to 140∘. Therefore, all the interior angles measure less than 180∘, so we conclude that this polygon is convex.

For the remainder of this explainer, we will be working with convex polygons.

Our next step is to derive a formula for π‘†οŠ, the sum of the interior angle measures of a polygon with 𝑛 sides.

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

For polygons with more than three sides, to find the sum of the interior angles, we split the polygon into triangles as shown below.

We know that the measures of the interior angles in each of the separate triangles must sum to 180∘.

In addition, the number of triangles within an 𝑛-sided polygon is 2 less than its number of sides, that is, π‘›βˆ’2. We can see that this is true for the polygon above, which has 5 sides and 5βˆ’2=3 triangles. It is easy to draw further examples to check this.

Therefore, the sum of the interior angles of a polygon with 𝑛 sides will be equal to (π‘›βˆ’2)180.

Formula: The Sum of the Measures of the Interior Angles of a Polygon

The sum, π‘†οŠ, of the measures of the interior angles of a polygon with 𝑛 sides is given by the formula 𝑆=(π‘›βˆ’2)180.

Let us now look at some examples of applying this formula.

Example 2: Finding the Sum of the Interior Angles of a Hexagon

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

Answer

Recall that π‘†οŠ, the sum of the interior angle measures of a polygon with 𝑛 sides, is given by the formula 𝑆=(π‘›βˆ’2)180.

A hexagon is a polygon with six sides and six vertices, so in this case, we have 𝑛=6.

Hence, we can substitute this value into the formula and simplify to find the sum of the interior angle measures: 𝑆=(6βˆ’2)Γ—180=4Γ—180=720.∘

We conclude that the sum of the interior angles of a hexagon is 720∘.

As the formula enables us to calculate the sum of the interior angle measures of an 𝑛-sided polygon, this means that if we are given an 𝑛-sided polygon and π‘›βˆ’1 of its interior angle measures, we can always work backward from the formula to find the missing angle. Here is an example.

Example 3: Finding the Measure of an 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

Recall that π‘†οŠ, the sum of the interior angle measures of a polygon with 𝑛 sides, is given by the formula 𝑆=(π‘›βˆ’2)180.

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

Using information from the question, we can substitute for all of the interior angle measures except π‘šβˆ π· to give 𝑆=137+78+113+π‘šβˆ π·+131=459++π‘šβˆ π·.∘∘∘∘∘

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

We have now obtained two separate equations for π‘†οŠ« and can equate them to get 459+π‘šβˆ π·=540.∘∘

Finally, we solve for π‘šβˆ π· by subtracting 459 from both sides, so π‘šβˆ π·=540βˆ’459=81.∘∘∘

We have found that π‘šβˆ π·=81∘.

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

Definition: Regular and Irregular Polygons

A polygon is considered to be regular when all its angles are of equal measure and all its sides are equal in length. 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 each interior angle, 𝐴, by dividing 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 or equiangular polygon, since all 𝑛 angles are equal in measure.

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

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

Formula: Interior Angles of a Regular Polygon

The measure, 𝐴, of each interior angle of a regular 𝑛-sided polygon is given by the formula 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 we should make sure that the shape in question is indeed a regular polygon.

This fact can be seen by imagining a regular arrangement of angle measures 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 angle measures may be greater than 180∘, but this is outside the scope of this explainer.

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

Let us now look at a question that uses 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 regular polygon is 179∘. How many sides does it have?

Answer

Recall that the measure, 𝐴, of each interior angle of a regular 𝑛-sided polygon is given by the formula 𝐴=(π‘›βˆ’2)180𝑛.

For this question, we are 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=179 into the formula for each interior angle in a regular polygon, giving 179=(π‘›βˆ’2)180𝑛.

We can now solve this equation by first multiplying both sides by 𝑛 and then grouping and simplifying the terms as follows: 179𝑛=(π‘›βˆ’2)180179𝑛=180π‘›βˆ’360.

Adding 360 to both sides, we get 179𝑛+360=180𝑛, and subtracting 179𝑛 from both sides gives 360=𝑛, which is the same as 𝑛=360.

We have found that the number of sides (and angles) in this shape is 360.

Even if the above question had not mentioned the regularity of the polygon, we could have still applied the same formula. This is because the measure of each interior angle of a regular 𝑛-sided polygon is the same as the measure of each interior angle of an equiangular 𝑛-sided polygon. However, it is unusual to meet examples of this type.

Next, we look at the exterior angles. An interesting property worth noting is that the exterior angle measures of a polygon sum to 360∘.

To show this, we give a sketch for the simplest polygon, which is a triangle (𝑛=3). We can imagine a general triangle with interior angle measures π‘ŽοŠ§, π‘ŽοŠ¨, and π‘ŽοŠ©. By extending each side beyond the vertices, we can form the exterior angles with measures π‘οŠ§, π‘οŠ¨, and π‘οŠ©.

Now, consider 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 exterior angles. Upon arriving back at point 𝑃, the arrow will have completed one complete turn of 360∘, so 𝑏+𝑏+𝑏=360.∘

This property of exterior angles is particularly useful when answering questions about regular polygons, as we will see in our next example.

Example 5: Finding the Number of Sides of a Regular Polygon given an Exterior Angle

If a regular polygon has an exterior angle of 90∘, find the number of sides it has.

Answer

Recall that the measures of the exterior angles of a polygon sum to 360∘.

In a regular 𝑛-sided convex polygon, there will be 𝑛 exterior angles, all of which must have the same measure. Therefore, we deduce that 𝑛×=360.exteriorangle

As we are told that the regular polygon has an exterior angle measure of 90∘, then substituting this value, we get 𝑛×90=360.

Finally, dividing both sides by 90 gives 𝑛=4.

Notice that we could have worked out our answer using an alternative method, as follows. Recall that at any given vertex, the measures of the interior and exterior angles sum to 180∘; that is, interiorangleexteriorangle+=180.

Substituting the value of the exterior angle and then rearranging, we get interiorangleinteriorangle+90=180=180βˆ’90=90.

Now, we know that the measure of each interior angle of a regular 𝑛-sided polygon is given by the expression (π‘›βˆ’2)180𝑛, so we have 90=(π‘›βˆ’2)180𝑛.

Multiplying both sides by 𝑛 gives 90𝑛=(π‘›βˆ’2)180, and distributing over the parentheses on the right-hand side, we get 90𝑛=180π‘›βˆ’360.

Then, subtracting 180𝑛 from both sides and dividing through by βˆ’90, we obtain 𝑛=4, as before.

We conclude that the regular polygon has 4 sides, which means it must be a square.

Next, we will show how to find the measure of an exterior angle of a regular polygon given the number of sides it has.

Example 6: Finding the Measures of the Interior and Exterior Angles of a Regular Polygon

Find π‘₯ and 𝑦.

Answer

The diagram shows a regular 8-sided polygon (i.e., an octagon) with an exterior angle measure of π‘₯ and an interior angle measure of 𝑦.

Recall that the measures of the exterior angles of a polygon sum to 360∘.

Starting with the exterior angles of a regular octagon, there are 8 of these, all of which must have the same measure, π‘₯. This implies that 8π‘₯=360, and dividing both sides by 8 gives π‘₯=45∘.

Recall also that at any given vertex, the measures of the interior and exterior angles sum to 180∘. Therefore, in this case, we must have π‘₯+𝑦=180.

Substituting π‘₯=45 from above and rearranging, we get that the interior angle measure is 𝑦=180βˆ’45=135.∘

We have found that π‘₯=45∘ and 𝑦=135∘.

In our final example, we will need to work back from the sum of the measures of the interior angles in a polygon to find the number of sides.

Example 7: Finding the Number of Sides of a Polygon given the Measures of Its Interior Angles

If the measures of two interior angles of a polygon are 120∘ and 40∘ and the sum of the rest of the angles is 380∘, find the number of sides.

Answer

Recall that the sum, π‘†οŠ, of the interior angle measures of a polygon with 𝑛 sides is given by the formula 𝑆=(π‘›βˆ’2)180.

Here, we are given the measures of two of the interior angles, together with the sum of the rest. Therefore, we can sum these angles to give π‘†οŠ, where 𝑛 is a number yet to be determined: 𝑆=120+40+380=540.

Now that we know the value of π‘†οŠ, we can substitute it into the formula to get 540=(π‘›βˆ’2)180.

Lastly, we solve this equation for 𝑛. Distributing over the parentheses on the right-hand side, we have 540=180π‘›βˆ’360. and adding 360 to both sides gives 900=180𝑛.

Dividing through by 180, we get 𝑛=5.

Thus, we have found that the polygon has 5 sides, which means it is a pentagon.

Let us finish by recapping some key concepts from this explainer.

Key Points

  • A convex polygon is a polygon with all interior angles measuring less than 180∘. A concave polygon is a polygon with one or more interior angles measuring more than 180∘.
  • The sum, π‘†οŠ, of the interior angle measures of a polygon with 𝑛 sides is given by the formula 𝑆=(π‘›βˆ’2)180.
  • A polygon is considered to be regular when all its angles have equal measures and all its sides are equal in length. In any other case, the polygon is considered to be irregular.
  • The measure, 𝐴, of each interior angle of a regular 𝑛-sided polygon is given by the formula 𝐴=(π‘›βˆ’2)180𝑛.
  • The interior angle measures of a regular polygon will always be less than 180∘, whereas the interior angle measures of an irregular polygon may be greater than 180∘.
  • The sum of the exterior angle measures of a polygon is 360∘.

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