Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![**Topic: Interior Angles in n-gons**
3. Two equations about interior angles in \( n \)-gons are given here.
**a. Explain what each one calculates.**
1. \((n - 2) \cdot 180^\circ\)
2. \(\frac{(n - 2) \cdot 180^\circ}{n}\)
### Explanation:
1. **Total Sum of Interior Angles:**
The equation \((n - 2) \cdot 180^\circ\) calculates the **total sum of the interior angles** in an \( n \)-gon, where \( n \) represents the number of sides. This formula is derived from the fact that any \( n \)-gon can be divided into \((n - 2)\) triangles, and each triangle has an angle sum of \( 180^\circ \).
2. **Measure of Each Interior Angle in a Regular n-gon:**
The equation \(\frac{(n - 2) \cdot 180^\circ}{n}\) calculates the **measure of each interior angle** in a regular \( n \)-gon (a polygon with all sides and angles equal). This is done by dividing the total sum of the interior angles by the number of angles, which is \( n \).
These equations are essential for understanding the geometric properties of polygons and are widely used in various mathematical and practical applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c2f64fc-e07a-43b3-a4b0-6f03d401f48c%2F1187653e-f227-489e-b6eb-27e27d315f42%2Feqxbig_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Interior Angles in n-gons**
3. Two equations about interior angles in \( n \)-gons are given here.
**a. Explain what each one calculates.**
1. \((n - 2) \cdot 180^\circ\)
2. \(\frac{(n - 2) \cdot 180^\circ}{n}\)
### Explanation:
1. **Total Sum of Interior Angles:**
The equation \((n - 2) \cdot 180^\circ\) calculates the **total sum of the interior angles** in an \( n \)-gon, where \( n \) represents the number of sides. This formula is derived from the fact that any \( n \)-gon can be divided into \((n - 2)\) triangles, and each triangle has an angle sum of \( 180^\circ \).
2. **Measure of Each Interior Angle in a Regular n-gon:**
The equation \(\frac{(n - 2) \cdot 180^\circ}{n}\) calculates the **measure of each interior angle** in a regular \( n \)-gon (a polygon with all sides and angles equal). This is done by dividing the total sum of the interior angles by the number of angles, which is \( n \).
These equations are essential for understanding the geometric properties of polygons and are widely used in various mathematical and practical applications.
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