The sum of interior angles of any polygon can be calculated using a formula. The formula is derived considering that we can divide any polygon into triangles. If the polygon is regular, we can calculate the measure of one of its interior angles by dividing the total sum by the number of sides of the polygon.

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

## Sum of interior angles of a polygon

We can find the sum of interior angles of any polygon using the following formula:

$latex (n-2)\times 180$° |

where *n* is the number of sides of the polygon. For example, we use $latex n = 5$ for a pentagon.

This formula works regardless of whether the polygon is regular or irregular. This is because a polygon always maintains the same sum of interior angles.

Let’s look at some examples. A square has four sides, so we have $latex n = 4$. When we use this in the formula, we have:

$latex (n-2)\times 180$°

$latex =(4-2)\times 180$°

$latex =(2)\times 180$°

$latex =360$°

Now, if we consider a hexagon, which has six sides, we have:

$latex (n-2)\times 180$°

$latex =(6-2)\times 180$°

$latex =(4)\times 180$°

$latex =720$°

The following is a table with the sum of the interior angles of the most common polygons:

Polygon | Number of sides | Sum of angles |

Triangle | 3 | 180° |

Quadrilateral | 4 | 360° |

Pentagon | 5 | 540° |

Hexagon | 6 | 720° |

Heptagon | 7 | 900° |

Octagon | 8 | 1080° |

Nonagon | 9 | 1260° |

Decagon | 10 | 1440° |

## Internal angles of a regular polygon

We can determine the measure of each of the interior angles of a regular polygon starting from the sum of all the interior angles. We know that a regular polygon has all its sides with the same length and all its angles with the same measure.

Therefore, we use the sum of the interior angles of a polygon and divide it by the number of sides of the regular polygon to find the measure of each angle. Using this, we have the following formula:

$latex \frac{(n-2)\times 180}{n}$ |

where *n* is the number of sides of the regular polygon. For example, a regular heptagon has 7 sides.

Let’s look at some examples. For a square, we use $latex n = 4$. Earlier, we saw that the sum of interior angles in a square equals 360°. Therefore, when we divide this by 4, we have:

360 ° ÷ 4 = 90 °

Each internal angle of a square measures 90°.

Now, in the case of a hexagon, we saw that the sum of its internal angles is equal to 720°. Therefore, by dividing by 6, which is the number of sides of the hexagon, we have:

720°÷6 = 120°

Each internal angle of a hexagon measures 120°.

The following is a table of common regular polygon interior angle measures:

Polygon | Each angle |

Triangle | 60° |

Square | 90° |

Pentagon | 108° |

Hexagon | 120° |

Heptagon | 128.57° |

Octagon | 135° |

Nonagon | 140° |

Decagon | 144° |

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## Proof of the formula for interior angles

Let’s consider the following polygon that has the vertices $latex V_{1}$ up to $latex V_{n}$.

If we join $latex V_{1}$ to each vertex except for $latex V_{2}$ and $latex V_ {n}$, we can form $latex (n-2)$ triangles, where, *n* is the number of sides of the polygon.

Now, we know that the sum of interior angles in a triangle is always equal to 180°. Therefore, the sum of interior angles of a polygon with *n* sides is equal to $latex (n-2) \times 180$°.

## Examples of interior angles of polygons

### EXAMPLE 1

What is the sum of the interior angles of an 11-sided polygon?

**Solution:** We have to use the formula for the sum of interior angles with $latex n = 11$. Therefore, we have:

$latex (n-2)\times 180$°

$latex =(11-2)\times 180$°

$latex =(9)\times 180$°

$latex =1620$°

The sum of interior angles of an 11-sided polygon is equal to 1620°.

### EXAMPLE 2

Determine the measure of the interior angles of a regular 11-sided polygon.

**Solution:** Since the polygon is regular, we can use the sum obtained in the previous example and divide by 11 since all the angles are equal. Therefore, we have:

1620°÷11≈147.27°

Each internal angle in an 11-sided regular polygon measures 147.27°.

## See also

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