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Chapter 24: Chapter 24

Problem 493

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What is the measure of the interior angle of a regular triangle? A regular quadrilateral? A regular 10 -gon? A regular 2000 -gon?

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The interior angle measures of the given regular polygons are as follows: Regular triangle (3 sides) - \(60°\); Regular quadrilateral (4 sides) - \(90°\); Regular 10-gon (10 sides) - \(144°\); Regular 2000-gon (2000 sides) - approximately \(179.64°\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Regular Triangle (n=3)

To find the interior angle measure of a regular triangle, we will use the formula: \(Interior \ angle \ measure = \frac{(n-2) × 180°}{n}\) \(= \frac{(3-2) × 180°}{3}\) \(= \frac{1 × 180°}{3}\) \(= 60°\) So, the measure of the interior angle of a regular triangle is 60°.

Step 2: Regular Quadrilateral (n=4)

To find the interior angle measure of a regular quadrilateral, we will use the formula: \(Interior \ angle \ measure = \frac{(n-2) × 180°}{n}\) \(= \frac{(4-2) × 180°}{4}\) \(= \frac{2 × 180°}{4}\) \(= 90°\) So, the measure of the interior angle of a regular quadrilateral is 90°.

Step 3: Regular 10-gon (n=10)

To find the interior angle measure of a regular 10-gon, we will use the formula: \(Interior \ angle \ measure = \frac{(n-2) × 180°}{n}\) \(= \frac{(10-2) × 180°}{10}\) \(= \frac{8 × 180°}{10}\) \(= 144°\) So, the measure of the interior angle of a regular 10-gon is 144°.

Step 4: Regular 2000-gon (n=2000)

To find the interior angle measure of a regular 2000-gon, we will use the formula: \(Interior \ angle \ measure = \frac{(n-2) × 180°}{n}\) \(= \frac{(2000-2) × 180°}{2000}\) \(= \frac{1998 × 180°}{2000}\) \(= 179.64°\) So, the measure of the interior angle of a regular 2000-gon is approximately 179.64°.

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Most popular questions from this chapter

Chapter 24

Show that the measure of the exterior angle of a regular n-sided polygon is given by the formula \(360 / \mathrm{n}\).
Chapter 24

What relation holds for an angle of a polygon and its exterior angle?
Chapter 24

The interior angles of a polygon are in arithmetic progression; the least angle is \(120^{\circ}\) and the common difference is \(5^{0}\). Find the number of sides of the polygon.
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\(\underline{A X}\) and \(\underline{B X}\) are two adjacent sides of a regular polygon. If the measure of angle \(\mathrm{ABX}\) equals \(1 / 3\) the measure of angle \(\mathrm{AXB}\), how many sides has the regular polygon?
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Find the number of degrees in the measure of each exterior angle of a regular polygon which has 12 sides.
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