# Angles in Polygons

## Understanding the Sum of Angles in Polygons

Have you ever wondered about the total of angles in a polygon? Perhaps you have learned that triangles have an angle sum of 180 degrees, and quadrilaterals have 360 degrees. But what about polygons with more sides? In this article, we will delve into the world of angles in polygons and uncover the formula for finding their sum.

## What Exactly is a Polygon?

The term “poly” means many, so a polygon simply refers to a shape with multiple sides. In mathematics, we define “many” as three or more, meaning that a polygon can be any 2D shape that is not a circle. A regular polygon has all sides and angles equal.

## Interior Angles in Polygons

When discussing angles in a polygon, we are referring to the total of its interior angles. It is essential to understand this term as we will be using it throughout this article. An interior angle is an angle inside the polygon (as shown in the diagram above). The sum of interior angles represents the combined value of all the angles within the polygon. For a triangle, the sum of its interior angles is 180 degrees, and for a quadrilateral, it is 360 degrees.

Formula for the Sum of Interior Angles:

• For any polygon with n sides, the sum of its interior angles is (n-2) * 180 degrees.

This formula enables us to calculate the sum of interior angles for any polygon. For instance, a pentagon has five sides, giving it a sum of interior angles of (5-2)*180 = 540 degrees. A nonagon, on the other hand, with nine sides, has a sum of interior angles of (9-2)*180 = 1260 degrees.

Let's apply this formula to a more complex shape, such as the 14-sided polygon in the image above. Using the formula, we can calculate its sum of interior angles to be (14-2)*180 = 2160 degrees.

To determine the missing angle in a shape, we can make use of the formula and basic mathematical operations. For example, in the image of the quadrilateral above, we know that the sum of interior angles is 540 degrees, and two angles are already given at 90 degrees each. Therefore, we can calculate the missing angle by subtracting the two given angles from 540 degrees, resulting in 360 degrees.

## Table of Sum of Interior Angles

The table below displays the sum of interior angles for the first eight polygons. You can also verify these results for yourself using the formula.

Number of SidesSum of Interior Angles3 (triangle)180 degrees4 (quadrilateral)360 degrees5 (pentagon)540 degrees6 (hexagon)720 degrees7 (heptagon)900 degrees8 (octagon)1080 degrees

## Understanding Regular Polygons

A regular polygon is a polygon with equal sides and angles. We can apply the formula to determine the interior angle of a regular polygon by dividing the sum of interior angles by the number of sides. For example, a regular hexagon has a sum of interior angles of 720 degrees, making each angle 720/6 = 120 degrees.

Let's apply this to a real-life scenario. In the image above, we see a tiling pattern composed of three regular pentagons. By using the formula and basic mathematical operations, we can calculate the angle labeled “x”. The sum of interior angles for each regular pentagon is (5-2)*180 = 540 degrees. Therefore, each interior angle is 540/5 = 108 degrees. We can then utilize the fact that angles around a point sum up to 360 degrees to find x by subtracting the given angles from 360, resulting in 360 - (3*108) = 36 degrees.

## Exploring Exterior Angles in Polygons

Besides interior angles, there are also exterior angles for each interior angle in a polygon. However, we will delve further into exterior angles and their properties in another article.

With a better understanding of angles in polygons, you can apply this knowledge to tackle more complex problems and calculations. Always remember to use the formula (n-2)*180 when finding the sum of interior angles for a polygon. Keep exploring and learning about polygons to deepen your understanding of geometry.

## The Basics of Angles in Polygons

Are you familiar with exterior angles? These angles are formed when a straight line is drawn outside of a polygon, intersecting with any side of the shape. While this concept may seem daunting, a visual representation can make it easier to understand.

In the diagram below, the orange angles represent the interior angles of the polygon, while the green angles are the exterior angles. Since both types of angles lie on the same straight line, their sum is always 180 degrees. Therefore, to find the measure of an exterior angle, simply subtract the measure of its corresponding interior angle from 180 degrees.

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