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Chapter 6: Problem 20

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

Short Answer

Expert verified
Each central angle of a square is 90 degrees.

Step by step solution

01

- Identify the number of sides in a square

A square is a regular polygon with 4 sides.
02

- Understand the formula for the central angle

The measure of each central angle in a regular polygon can be found using the formula \[ \text{Central Angle} = \frac{360^{\circ}}{n} \] where \( n \) is the number of sides in the polygon.
03

- Plug in the number of sides

Substitute the number of sides (\( n = 4 \)) into the formula: \[ \text{Central Angle} = \frac{360^{\circ}}{4} \]
04

- Perform the division

Calculate the value: \[ \text{Central Angle} = 90^{\circ} \]

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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 shape with all sides and angles equal. Common examples of regular polygons include equilateral triangles, squares, and regular pentagons. These shapes have a high degree of symmetry, making them easy to study in geometry. Each angle and side is identical, which simplifies calculations.
Regular polygons can be distinguished from irregular polygons, which do not have equal sides and angles. In problems involving regular polygons, the uniformity of sides and angles plays a crucial role.

It is helpful to remember:
  • Equilateral triangles have 3 equal sides and angles.
  • Squares have 4 equal sides and angles.
  • Pentagons have 5 equal sides and angles, and so on.
Understanding these basics is essential for solving problems related to regular polygons.
Applying the Central Angle Formula
The central angle of a regular polygon is an angle formed at the center of the polygon by two adjacent sides. This angle helps in understanding the overall geometry of the shape.
To find the central angle of any regular polygon, we use the formula: \[\text{Central Angle} = \frac{360^{\text{掳}}}{n}\]
Where:
  • is the number of sides in the polygon
  • 360掳 represents the total degrees in a full circle
By dividing 360掳 by the number of sides, we get the measure of each central angle.
For instance, for a square ( = 4):\[\text{Central Angle} = \frac{360^{\text{掳}}}{4} = 90^{\text{掳}}\]This means each central angle of a square is 90掳, helping us visualize and solve further geometry problems.
Geometry Problem-Solving Using Central Angles
Solving geometry problems often requires breaking down the problem into smaller, manageable parts. One of the fundamental techniques is using properties like the central angle.
Knowing how to calculate central angles can help in:
  • Finding the measure of other angles in the polygon
  • Determining the length of each side when combined with other formulas
  • Understanding symmetry and rotational properties of the polygon
When faced with a geometry problem involving regular polygons, start by identifying and calculating the central angle. This can provide the foundation for more complex calculations, making problem-solving efficient and accurate.
Practicing different types of problems ensures you gain confidence. Geometry becomes easier as you get familiar with these core concepts.

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