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Geometry

The effects on the interior angles of polygons are observed when the number of sides and the positions of the corners of the polygons are changed.

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After completing this tutorial, you will be able to complete the following:

- Determine the sum of the interior angles of a polygon.
- Explain the relationship between the number of sides and the sum of the interior angles of a polygon.

A polygon is a closed figure made by joining line segments each of which intersects exactly two others.

A polygon is named based on the number of sides and angles it possesses. Think about the meaning of the following prefixes.

Regular polygons

Some of the polygons in this Activity Object are regular polygons. A regular polygon is one whose sides are all the same length, and whose angles are all equal. For example:

An angle is formed by two rays that share the same endpoint.

Angles are measured in degrees. We use a protractor to measure angles. Interior angles are those that are formed in the inside of a polygon. The sum of the interior angles for each polygon is consistent for all types of polygons whether they are regular or irregular, large or small. During the Activity Object, the learner moves the slider to change the number of sides of the polygon, and the sum of the interior angles is shown in a label along the bottom of the window. The below chart shows the sums of the interior angles of polygons.

If we study the difference between the sums of the interior angles as the number of sides of the polygon increases, a pattern emerges. Each time a side is added to the polygon the sum of the interior angles increases by 1800.

Example:

The difference between the sums of the interior angles of a pentagon and a quadrilateral can be shown in this number sentence.540° - 360° = 180°

An algebraic explanation for calculating the sum of the interior angles of a polygon.

Since the angles of a triangle have a sum of 180°, we can draw all the diagonals (segments that join non-adjacent vertices) that have an endpoint at A.

This divides the polygon into n - 2 triangles, where n represents the number of sides of the polygon.

The sum of all the angles of the polygon equals the sum of the angles of the triangles: 180° ? (n - 2). So, for the example above, the sum of the interior angles is:

If the polygon can be classified as regular, then you can use the formula can be used to find the measure of each interior angle. For a regular pentagon: .

Each interior angle of a regular pentagon measures 108°.

Approximate Time | 15 Minutes |

Pre-requisite Concepts | Students should know the definitions of angle and polygon. |

Course | Geometry |

Type of Tutorial | Skills Application |

Key Vocabulary | interior angles, polygons, position of corners |