Questions and Answers
What does the term 'n' represent in polygon formulas?
Number of angles/sides
How do you calculate the number of triangles formed in a polygon?
n-2
What is the formula for the sum of the interior angles of a polygon?
(n-2)180
How do you calculate one interior angle of a regular polygon?
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What is the sum of the exterior angles of a polygon?
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What is the formula for one exterior angle of a regular polygon?
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If a regular polygon has an interior angle of 162°, how many sides does it have?
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For a regular polygon with an exterior angle of 20°, how many sides does it have?
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What is the measure of each interior angle in a regular polygon with 15 sides?
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If the sum of the interior angles of a regular polygon is 3060°, how many sides does it have?
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If each interior angle of a regular polygon is eight times as large as its corresponding exterior angle, how many sides does the polygon have?
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Determine the sum of the measures of the angles in a 12-sided convex polygon.
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Study Notes
Polygon Basics
- A polygon's characteristics include the number of angles/sides represented by the variable n.
- For any polygon, the number of triangles formed is calculated as n - 2.
Angle Calculations
- The sum of the interior angles of a polygon can be determined using the formula (n - 2) × 180°.
- Each interior angle of a regular polygon is derived from the formula (\frac{(n - 2) \times 180}{n}).
Exterior Angles
- The total sum of exterior angles for any polygon is always 360°.
- One exterior angle of a regular polygon can be calculated with the formula (\frac{360}{n}).
Finding the Number of Sides
- To find the number of sides in a regular polygon given an interior angle, use the relation: (180 - \text{interior angle} = \text{exterior angle}) and then (360/\text{exterior angle} = n).
- For a polygon with a specific interior angle of 162°, the calculation results in 20 sides using the above formula.
Regular Polygon Examples
- In a regular polygon with 15 sides, the sum of interior angles is calculated as (15 - 2) × 180° = 2340°, leading to each angle being 156°.
- If the sum of interior angles is given (e.g., 3060°), use ((n-2) \times 180 = 3060) to deduce the number of sides (19 sides) and then find each exterior angle (approximately 18.95°).
Special Conditions
- The relationship where each interior angle is eight times its corresponding exterior angle leads to 18 sides, utilizing the equation (x + 8x = 180) for calculations.
Case Studies
- To find the total angle measures in a 12-sided convex polygon, apply (12 - 2) × 180° = 1800°.
- In a scenario where the interior angle is 162°, a multiple equation approach simplifies to find n = 18.
Summary of Formulas
- Number of sides (n) = angles + 2
- Interior angles sum = (n - 2) × 180°
- Each interior angle = (\frac{(n - 2) \times 180}{n})
- Each exterior angle = (\frac{360}{n})
These formulas and methods allow for determining the properties and dimensions of polygons efficiently.
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Test your knowledge of polygon formulas with these flashcards. Each card presents a specific term related to polygons, accompanied by its definition. Perfect for students looking to understand the properties and calculations involving polygons.
