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Chapter 10: Problem 59

Describe how to find the measure of an angle of a regular pentagon.

Short Answer

Expert verified
The measure of an angle in a regular pentagon is 108 degrees.

Step by step solution

01

Compute the total sum of all interior angles in the polygon

In any polygon, the total sum of all interior angles is given by the formula \((n-2) \times 180\) degrees, where n is the number of sides. For a pentagon, n equals 5. So, the sum of all interior angles is \((5-2) \times 180 = 540\) degrees.
02

Find the measure of one angle

In a regular pentagon, all angles are equal. Therefore, the measure of one angle can be found by dividing the total sum of the interior angles by the number of angles, which is 5 in this case. So, the measure of one angle equals \(540 梅 5 = 108\) degrees.

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Key Concepts

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

Interior Angles
The interior angles of a polygon are the angles found inside the shape, formed by its sides. These angles are one of the essential parts in understanding the structure and properties of polygons.
The sum of the interior angles of a polygon depends on the number of its sides. To calculate this sum, we use the formula \((n-2) \times 180\), where \(n\) represents the number of sides.
For example, in a pentagon, which has five sides, the sum of the interior angles can be calculated as \((5-2) \times 180 = 540\) degrees.
  • Key Idea: The more sides a polygon has, the larger the sum of its interior angles will be.
  • For irregular polygons, the angles may differ, but this formula will still give you the total sum.
Regular Polygon
A regular polygon is a type of polygon where all its sides and angles are equal. This uniformity helps simplify calculations for angles and other properties.
Because each interior angle is the same in a regular polygon, once you know the sum of all the angles, figuring out an individual angle becomes a straightforward division problem.
Regular polygons include familiar shapes like squares (4 sides), hexagons (6 sides), and pentagons (5 sides).
  • Key Feature: Regular polygons have symmetry, which can be useful in various geometric applications.
  • Each angle in a regular polygon can be calculated once the sum of the interior angles is known.
Pentagon
A pentagon is a polygon with five sides and five angles. It can be regular, with all sides and angles equal, or irregular, with varying lengths and angles.
For a regular pentagon, because all the angles are the same, every angle measure can be easily derived once the total interior angle sum is known.
Each angle in a regular pentagon is calculated to be \(108\) degrees, derived from dividing the total sum of \(540\) degrees by 5.
  • Regular pentagons appear in nature and design due to their symmetry and balance.
  • Understanding pentagons is a step towards exploring more complex polygons.
Angle Measure
Angle measurement is crucial in geometry for understanding the size and shape of different geometric figures. When dealing with regular polygons, measuring an individual angle involves dividing the total sum of interior angles by the number of angles.
For a regular polygon like a pentagon, this calculation provides precise and consistent angle measures.
In our example of a regular pentagon, the single angle measure is \(108\) degrees, which simplifies understanding and working with such shapes in geometric problems.
  • Having a consistent angle measure in regular polygons aids in creating designs and solving geometrical problems efficiently.
  • Angle measurement concepts are foundational and are connected to other geometric and mathematical explorations.

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