The total sum of all the interior angles of a hexagon equals 720°. The sum always holds regardless of whether the hexagon is regular or irregular. The measure of each individual angle in a regular hexagon can be determined by dividing the sum by 6. However, for irregular hexagons, we have to use the measures of other angles to determine the measure of a missing angle.

Here, we will learn more about the interior angles of hexagons.

## Sum of interior angles of a hexagon

The sum of all the interior angles of a hexagon is always equal to 720°. This is true regardless of whether the hexagon is regular or irregular. The sum of angles is obtained using the formula for the sum of polygons angles:

$latex (n-2)\times 180$° |

where, *n* is the number of sides of the polygon. For a hexagon, we use $latex n = 6$. Therefore, we have:

$latex (n-2)\times 180$°

$latex =(6-2)\times 180$°

$latex =(4)\times 180$°

$latex =720$°

The formula for the sum of interior angles can be obtained when we divide any polygon into triangles as in the following diagram:

We know that the sum of interior angles of any triangle is equal to 180°. Furthermore, we can form a total of *n*-2 triangles in any polygon, where, *n* is the number of sides of the polygon. This means that $latex (n-2) \times 180$° corresponds to the sum of all the interior angles of the polygon.

## How to determine the measures of all interior angles of hexagons?

When we have a regular hexagon, we can determine the measures of each interior angle by dividing the total sum of angles by 6. This is possible because all interior angles in a regular polygon have the same measure. Therefore, we have:

720°÷6 = 120°

Each interior angle in a regular hexagon has a measure equal to 120°.

To determine the measures of any missing angle in irregular hexagons, we need to know the measures of the other angles since these hexagons have internal angles that are different from each other.

For example, if we have a hexagon with interior angles 100°, 110°, 120°, 130°, 150°, we can determine the measure of the sixth angle by adding the known angles and subtracting from 720°:

100°+110°+120°+130°+150° = 610°

⇒ 720°-610° = 110°

The measure of the missing angle is 110°.

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## Examples of interior angles of hexagons

### EXAMPLE 1

Find the missing angles in the hexagon below.

**Solution:** We have the measures of five of the six interior angles of the irregular hexagon. We know that the total sum equals 720°, so we can add the known angles and subtract from 720°:

110°+140°+100°+120°+150° = 620°

⇒ 720°-620° = 100°

The measure of the missing angle is 100°.

### EXAMPLE 2

Find the missing angles in the irregular hexagon below.

**Solution:** Interior angles that are represented by the same number of lines and that share the same color are the same. This means that angle *a* has a measure of 130°.

Also, we know that angles *b* and *c* have the same measure. We can find the measures of these angles by adding the known angles and subtracting from 720°:

100°+130°+130°+120° = 480°

⇒ 720° -480° = 240°

The two missing angles are equal, so we divide the result by 2 to get the measure of each. Therefore, we have *b*=120° and *c*=120°.

## See also

Interested in learning more about the interior angles of a polygon? Take a look at these pages:

- Interior Angles of a Hexagon – Formula and Examples
- Exterior Angles of a Hexagon – Formula and Examples
- Interior Angles of a Heptagon – Formula and Examples
- Interior Angles of a Trapezoid – Formula and Examples
- Sum of Interior Angles of a Polygon – Formula and Examples
- Interior and Exterior Angles of a Polygon

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