Hexagons have a sum of interior angles of 720°. A regular hexagon has all its angles with the same measure, so each interior angle measures 120°. On the other hand, irregular hexagons have angles with different measures, but they always have a total sum equal to 720°.
Here, we will learn about the interior angles of hexagons in more detail. We will learn the formula used to find the sum of these angles and we will look at some examples.
Interior angles in a regular hexagon
A regular hexagon has the main characteristic that all its sides have the same length, all the sides of the hexagon are congruent. This also means that all interior angles have the same measure.
The sum of the interior angles of any hexagon is always equal to 720°. We can find the measure of each interior angle of a regular hexagon by dividing 720° by 6. Therefore, we have:
720°÷6 = 120°
Each interior angle in a regular hexagon measures 120°.
In the following diagram, we have a regular hexagon, which has sides of the same length and angles of the same measure. We can verify that by adding the six 120° angles, we get a total of 720°.
Formula to find the angles in a hexagon
We can find the sum of the interior angles of any polygon by applying the following formula:
In this formula, n is equal to the number of sides of the polygon. In this case, we use for a hexagon. Using this value, we have °. This shows that the sum of the interior angles of a hexagon is equal to 720°.
How to find a missing angle in a hexagon?
A missing angle of an irregular hexagon can be found by adding the measures of all known angles and subtracting the obtained value from 720°.
For example, in the following hexagon, we have the angles 110°, 140°, 100°, 120°, 150°.
We add all the known angles:
110°+140°+100°+120°+150° = 620°
Now, we subtract the obtained value from 720°:
720°-620° = 100°
The missing angle of the hexagon has a measure of 100°.
In the following example, we have to find more than one missing angle. In this case, the angles that are different have different symbols. For example, angles that have a double line are equal.
We know that the angles that have a double line are equal, therefore we have a=130°.
Similarly, angles b and c are equal since they have triple lines. We can find the measure of these angles by following a process similar to the previous example. We start by adding the known angles:
100°+130°+130°+120° = 480°
Now, we have to subtract this value from 480° to find the missing angles:
720°-480° = 240°
The missing angles have the same measure, so we divide the obtained value by 2 to obtain the value of each angle. Therefore, we have b=120° and c=120°.
Solved examples of interior angles of a hexagon
Determine the missing angle in the hexagon below.
Solution: We can look at the angles 130°, 90°, 115°, 140°, and 150°. Therefore, we start by adding them:
130°+90°+115°+140°+150° = 625°
Now, we subtract the obtained value from 720° to find the missing angle:
720°-615° = 95°
The missing angle measures 95°.
Find the missing angles in the hexagon below.
Solution: The angles shared by the three lines are equal. Therefore, we have a=120°.
Angles b and c also have the same measure since they are both represented by a double line. We can determine its measure by starting by adding the angles we know:
Now, we subtract the obtained value from 720° and we have:
720°-460° = 260°
The result represents the sum of the two missing angles. Since both angles are equal, we have to divide by 2 to get the value of one angle. Therefore, we have b = 130° and c = 130°.
Interested in learning more about the interior angles of a polygon? Take a look at these pages: