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Chapter 10: Problem 32

Find the measure of each exterior angle of each regular polygon. regular octagon

Short Answer

Expert verified
Each exterior angle of a regular octagon is 45掳.

Step by step solution

01

Understanding Exterior Angles

Exterior angles of any polygon are the angles formed between any side of the polygon and an extended adjacent side. In a regular polygon, all exterior angles are equal.
02

Formula for Exterior Angle

To find the measure of each exterior angle of a regular polygon, use the formula: \[ \text{Exterior angle} = \frac{360^\circ}{n} \]where \(n\) is the number of sides in the polygon.
03

Applying the Formula

For a regular octagon, the number of sides, \(n\), is 8. Substitute \(n = 8\) into the formula: \[ \text{Exterior angle} = \frac{360^\circ}{8} \]
04

Calculating the Exterior Angle

Perform the division: \[ \frac{360}{8} = 45^\circ \]Therefore, each exterior angle of a regular octagon is \(45^\circ\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Regular Polygons
A regular polygon is a unique type of geometric figure. It is characterized by having all sides of equal length and all interior angles of equal measure. This regularity gives these polygons a certain symmetry and uniformity that is aesthetically pleasing and makes calculations simpler. In this context, it's useful for simplifying the process of finding angles, both interior and exterior.
  • All sides are the same length.
  • All interior angles are equal.
  • Symmetry helps in calculating important properties like angles.

Understanding the nature of regular polygons is essential when working out problems involving exterior angles, as it provides a foundation for applying geometric formulas effectively.
Basic Concepts of Geometry
Geometry is a branch of mathematics that studies the sizes, shapes, and dimensions of different figures and spaces. It is an ancient discipline fundamental to understanding how shapes fit together and how to analyze their properties. Within geometry, you learn about different types of angles, lines, and polygons, each with their own rules and calculations.
In geometry, the concept of angles is critical:
  • Angles are measured in degrees.
  • Exterior angles are formed when a side of a polygon is extended.
  • The total of all exterior angles of any polygon is always 360 degrees.

These concepts provide a foundation that helps learners calculate more complex properties involving geometry, such as those found in regular polygons.
The Regular Octagon
An octagon is a polygon with eight sides and, in the case of a regular octagon, all sides and all angles are equal. This gives it a distinctive geometric charm. Specifically, for a regular octagon:
  • It has 8 sides: each of equal length.
  • It has 8 interior angles, each measuring 135 degrees.
  • It has 8 exterior angles, each measuring 45 degrees.

Understanding each of these characteristics helps in a variety of geometrical calculations, such as finding each exterior angle. For instance, using the formula \( \text{Exterior angle} = \frac{360^\circ}{n} \), where \( n \) is the number of sides, correctly yields 45 degrees for a regular octagon. This predictable nature of regular polygons, such as the regular octagon, makes such calculations straightforward.

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