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Chapter 5: Problem 2

What is the measure of each angle of an equiangular pentagon? An equiangular hexagon? Complete the table. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline \begin{array}{l} \text { Number of sides of } \\ \text { equiangular polygon } \end{array} & 5 & 6 & 7 & 8 & 9 & 10 & 12 & 16 & 100 \\ \hline \begin{array}{l} \text { Measure of each angle } \\ \text { of equiangular polygon } \end{array} & & & & & & & & & \\ \hline \end{array} $$

Short Answer

Expert verified
Each angle of an equiangular polygon with 5, 6, 7, 8, 9, 10, 12, 16, and 100 sides are 108掳, 120掳, 128.571掳, 135掳, 140掳, 144掳, 150掳, 157.5掳, and 176.4掳 respectively.

Step by step solution

01

Understanding Equiangular Polygons

An equiangular polygon is one where all the interior angles are equal. The measure of each interior angle of a regular (equiangular) polygon can be calculated using the formula: \[ \text{Measure of each angle} = \frac{(n-2) \times 180^{\circ}}{n} \] where \( n \) is the number of sides of the polygon.
02

Calculate Angle of Equiangular Pentagon (5 sides)

Substitute \( n = 5 \) in the formula: \[ \text{Measure of each angle} = \frac{(5-2) \times 180^{\circ}}{5} = \frac{3 \times 180^{\circ}}{5} = 108^{\circ} \]
03

Calculate Angle of Equiangular Hexagon (6 sides)

Substitute \( n = 6 \) in the formula: \[ \text{Measure of each angle} = \frac{(6-2) \times 180^{\circ}}{6} = \frac{4 \times 180^{\circ}}{6} = 120^{\circ} \]
04

General Calculation

Using the same formula, calculate the measure of each interior angle for polygons with 7, 8, 9, 10, 12, 16, and 100 sides: \( n = 7 \): \[ \text{Measure of each angle} = \frac{(7-2) \times 180^{\circ}}{7} = \frac{5 \times 180^{\circ}}{7} \approx 128.571^{\circ} \]\( n = 8 \): \[ \text{Measure of each angle} = \frac{(8-2) \times 180^{\circ}}{8} = \frac{6 \times 180^{\circ}}{8} = 135^{\circ} \]\( n = 9 \): \[ \text{Measure of each angle} = \frac{(9-2) \times 180^{\circ}}{9} = \frac{7 \times 180^{\circ}}{9} \approx 140^{\circ} \]\( n = 10 \): \[ \text{Measure of each angle} = \frac{(10-2) \times 180^{\circ}}{10} = \frac{8 \times 180^{\circ}}{10} = 144^{\circ} \]\( n = 12 \): \[ \text{Measure of each angle} = \frac{(12-2) \times 180^{\circ}}{12} = \frac{10 \times 180^{\circ}}{12} = 150^{\circ} \]\( n = 16 \): \[ \text{Measure of each angle} = \f rac{(16-2) \times 180^{\circ}}{16} = \frac{14 \times 180^{\circ}}{16} = 157.5^{\circ} \]\( n = 100 \): \[ \text{Measure of each angle} = \frac{(100-2) \times 180^{\circ}}{100} \approx 176.4^{\circ} \]
05

Complete the Table

Using the calculated angles, fill in the table as follows: \[\begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline \begin{array}{l} \text { Number of sides of } \ \text { equiangular polygon } \end{array} & 5 & 6 & 7 & 8 & 9 & 10 & 12 & 16 & 100 \ \hline \begin{array}{l} \text { Measure of each angle } \ \text { of equiangular polygon } \end{array} & 108^{\circ} & 120^{\circ} & 128.571^{\circ} & 135^{\circ} & 140^{\circ} & 144^{\circ} & 150^{\circ} & 157.5^{\circ} & 176.4^{\circ} \ \hline \end{array} \]

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Key Concepts

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

Angle Calculation
When working with polygons, calculating the measure of each interior angle is crucial. For an equiangular polygon, we use a specific formula: \( n \times (180 - (360/n)) \) where \( n \) represents the number of sides. This formula helps in understanding the internal geometry of any regular equiangular polygon. For example, to calculate the measure of each angle in an equiangular pentagon (5 sides), we substitute \( n = 5 \) into the formula: \[ \text{Measure of each angle} = \frac{(5-2) \times 180^{\circ}}{5} \approx 108^{\circ} \] Breaking it down makes it easier. Remember to follow each step and substitute correctly.
Geometry Formulas
Geometry involves several crucial formulas, especially for polygons. One such is the interior angle formula for regular polygons: \[ \text{Measure of each angle} = \frac{(n-2) \times 180^{\circ}}{n} \] This formula breaks down complex shapes into more understandable segments. Another essential concept is understanding that the sum of all interior angles in an \( n-\)sided polygon is \((n-2) \times 180掳 \). For instance, a hexagon (6 sides) would be calculated as: \[ \text{Sum of interior angles} = (6-2) \times 180^{\circ} = 720^{\circ} \] Divide this by 6 (the number of angles in a hexagon) to find each interior angle: \[ \text{Each angle} = \frac{720^{\circ}}{6} = 120^{\circ} \] With these formulas, tackling any polygon-related problem becomes manageable.
Polygon Properties
Polygons come with unique properties that distinguish them from other shapes. A polygon is a plane figure that is enclosed by straight lines. These lines are the sides or edges of the polygon. Polygons can be classified based on their sides and angles:
  • Regular Polygons: All sides and angles are equal (e.g., equilateral triangle, square).
  • Irregular Polygons: Sides and angles are not necessarily equal (e.g., scalene triangle).
Knowing these properties is essential. For example, if you understand what makes a polygon regular, calculating angles becomes straightforward using the standard formulas discussed earlier.
Regular Polygons
Regular polygons are characterized by equal side lengths and equal angles. They are a special category under polygons and display pronounced symmetry. Some common examples include:
  • Equilateral Triangle: 3 sides, each angle measuring 60掳.
  • Square: 4 sides, each angle measuring 90掳.
  • Regular Pentagon: 5 sides, each angle measuring 108掳.
The distinguishing feature is understanding regularity鈥攚hich directly leads to symmetry in both sides and angles. Using the formula for interior angles further simplifies this understanding.
These regular polygons often serve as a foundation for more complex geometric concepts, demonstrating perfect symmetry and uniformity.
Interior Angles
Interior angles are an integral part of understanding polygons. They are the angles formed inside a polygon at each vertex. For regular polygons, these angles are equal, and you can find their measure using specific geometry formulas. For an \( n-\)sided regular polygon, the formula is: \[ \text{Measure of each interior angle} = \frac{(n-2) \times 180^{\circ}}{n} \] This helps you solve problems efficiently. For instance, a regular hexagon's angle is found by: \[ \text{Measure of each angle} = \frac{(6-2) \times 180^{\circ}}{6} = 120^{\circ} \] Interior angles provide essential insights into polygon structure, aiding in both academic problems and real-world applications, such as design and architecture.

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Most popular questions from this chapter

How many sides does an equiangular polygon have if each interior angle measures \(156^{\circ} ?\)

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