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 “vertices”).

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 -sided polygons for .

Number of Sides | Name |
---|---|

3 | Triangle |

4 | Quadrilateral |

5 | Pentangon |

6 | Hexagon |

7 | Heptagon (septagon) |

8 | Octagon |

9 | Nonagon |

10 | Decagon |

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 .

### Definition: Convex and Concave Polygons

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

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 .

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 , while a concave polygon is a polygon with one or more interior angles measuring more than .

From the diagram, we see that the interior angles of this polygon have measures that range from up to . Therefore, all the interior angles measure less than , 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 , the sum of the interior angle measures of a polygon with sides.

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

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 .

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

Therefore, the sum of the interior angles of a polygon with sides will be equal to .

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

The sum, , of the measures of the interior angles of a polygon with sides is given by the formula

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 , the sum of the interior angle measures of a polygon with sides, is given by the formula

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

Hence, we can substitute this value into the formula and simplify to find the sum of the interior angle measures:

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

As the formula enables us to calculate the sum of the interior angle measures of an -sided polygon, this means that if we are given an -sided polygon and 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 , , , and , find .

### Answer

Recall that , the sum of the interior angle measures of a polygon with sides, is given by the formula

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, , can be expressed as follows:

Using information from the question, we can substitute for all of the interior angle measures except to give

A pentagon is a polygon with five sides and five vertices, and we can therefore substitute into the formula:

We have now obtained two separate equations for and can equate them to get

Finally, we solve for by subtracting 459 from both sides, so

We have found that .

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 .

For a regular polygon, we can find the measure of each interior angle, , by dividing the sum of the interior angles, , by the number of angles:

Note that this is only true for a regular or equiangular polygon, since all angles are equal in measure.

We have already shown that the sum of the measures of the interior angles for any -sided polygon can be found using the formula

We can therefore substitute for to find the formula for the measure of each interior angle in a regular polygon (in terms of ):

### Formula: Interior Angles of a Regular Polygon

The measure, , of each interior angle of a regular -sided polygon is given by the formula

It is worth noting that the above relationship implies that the value of is necessarily less than 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 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 “interior angles”!

For an irregular polygon, interior angle measures may be greater than , but this is outside the scope of this explainer.

As a final note, two line segments that form an angle of cannot be considered as two sides of a polygon, since they cannot be distinguished from a single segment. For this reason, 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 . How many sides does it have?

### Answer

Recall that the measure, , of each interior angle of a regular -sided polygon is given by the formula

For this question, we are given the measure of each interior angle of a regular polygon and are asked to find the number of sides and, therefore, the number of angles .

In order to find , we can substitute the known value into the formula for each interior angle in a regular polygon, giving

We can now solve this equation by first multiplying both sides by and then grouping and simplifying the terms as follows:

Adding 360 to both sides, we get and subtracting from both sides gives which is the same as .

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 -sided polygon is the same as the measure of each interior angle of an equiangular -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 .

To show this, we give a sketch for the simplest polygon, which is a triangle . We can imagine a general triangle with interior angle measures , , and . By extending each side beyond the vertices, we can form the exterior angles with measures , , and .

Now, consider an arrow at point 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 , the arrow will have completed one complete turn of , so

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 , find the number of sides it has.

### Answer

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

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

As we are told that the regular polygon has an exterior angle measure of , then substituting this value, we get

Finally, dividing both sides by 90 gives .

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 ; that is,

Substituting the value of the exterior angle and then rearranging, we get

Now, we know that the measure of each interior angle of a regular -sided polygon is given by the expression , so we have

Multiplying both sides by gives and distributing over the parentheses on the right-hand side, we get

Then, subtracting from both sides and dividing through by , we obtain , 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 and .

### Answer

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

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

Starting with the exterior angles of a regular octagon, there are 8 of these, all of which must have the same measure, . This implies that and dividing both sides by 8 gives .

Recall also that at any given vertex, the measures of the interior and exterior angles sum to . Therefore, in this case, we must have

Substituting from above and rearranging, we get that the interior angle measure is

We have found that and .

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 and and the sum of the rest of the angles is , find the number of sides.

### Answer

Recall that the sum, , of the interior angle measures of a polygon with sides is given by the formula

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 , where is a number yet to be determined:

Now that we know the value of , we can substitute it into the formula to get

Lastly, we solve this equation for . Distributing over the parentheses on the right-hand side, we have and adding 360 to both sides gives

Dividing through by 180, we get .

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 . A concave polygon is a polygon with one or more interior angles measuring more than .
- The sum, , of the interior angle measures of a polygon with sides is given by the formula
- 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, , of each interior angle of a regular -sided polygon is given by the formula
- The interior angle measures of a regular polygon will always be less than , whereas the interior angle measures of an irregular polygon may be greater than .
- The sum of the exterior angle measures of a polygon is .