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Chapter 6: Problem 19

Find the measure of each central angle of the given regular polygon. equilateral triangle

Short Answer

Expert verified
Each central angle of an equilateral triangle is 120 degrees.

Step by step solution

01

- Understand what a central angle is

A central angle of a regular polygon is an angle formed by two radii of the polygon that go from the center to two consecutive vertices.
02

- Identify the type of regular polygon

The given regular polygon is an equilateral triangle.
03

- Determine the number of sides

An equilateral triangle has 3 sides.
04

- Use the formula for a central angle

The formula to find the measure of each central angle in a regular polygon is: \( \text{Central Angle} = \frac{360^{\circ}}{n} \), where \( n \) is the number of sides of the polygon.
05

- Calculate the central angle

Substitute \( n = 3 \) into the formula: \( \text{Central Angle} = \frac{360^{\circ}}{3} = 120^{\circ} \).

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Key Concepts

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

equilateral triangle
An equilateral triangle is a special type of triangle where all three sides are of equal length and all three angles are equal.
This uniformity makes calculations simpler.
In the context of a regular polygon, the equilateral triangle has some interesting properties:
  • All angles in an equilateral triangle are 60 degrees.
  • It is the simplest type of regular polygon with the least number of sides.

These properties of equilateral triangles make them an excellent starting point for understanding more complex polygons.
polygon sides
A polygon is a closed figure with straight sides.
The number of sides in a polygon plays a crucial role in various calculations.
For example, an equilateral triangle has 3 sides, a square has 4 sides, and a pentagon has 5 sides.
In the case of the equilateral triangle, its 3 equal sides directly influence the angles within the polygon.
When dealing with any polygon, one must always count the number of sides accurately to use them in relevant formulas, such as for calculating angles.
angle calculation
To find the measure of each central angle in a regular polygon, we use a reliable formula: \( \text{Central Angle} = \frac{360^{\circ}}{n} \), where \( n \) is the number of sides.
With an equilateral triangle, substituting \( n = 3 \) into the formula gives us:
\[ \text{Central Angle} = \frac{360^{\circ}}{3} = 120^{\circ} \]
This formula works because a circle has 360 degrees, and for a regular polygon, it is evenly divided by the number of sides.
Hence, each central angle turns out to be 120 degrees for an equilateral triangle.
This principle applies to any regular polygon to find central angles, simplifying complex geometrical computations.

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Most popular questions from this chapter

Find the measure of each central angle of the given regular polygon. regular decagon

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