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Exploring Angles in Polygons
Aug 22, 2024
Understanding Interior and Exterior Angles in Polygons
Sum of Interior Angles
Formula:
(n - 2) * 180
n
= number of sides (or angles) in the polygon
Explanation:
Example with a polygon having 5 sides (pentagon):
Can be divided into 3 triangles
Sum of angles in triangles = 3 * 180 = 540 degrees
Therefore, sum of interior angles for a polygon with n sides = (n - 2) * 180
Measure of One Interior Angle
Formula:
(n - 2) * 180 / n
Application:
Used for regular polygons (equal angles and equal side lengths)
Gives the measure of one interior angle by dividing total interior angle sum by number of angles.
Measure of One Exterior Angle
Formula:
360 / n
Application:
Used for regular polygons to find one exterior angle.
Sum of Exterior Angles
Always add up to
360 degrees
regardless of polygon type (regular or irregular).
Visualization: Formed angles on the outside of the polygon when extended add up to 360 degrees (like a circle).
Example Problems
Example 1: Finding an Interior Angle
Given a polygon with 5 sides:
Calculate sum of angles:
(5 - 2) * 180 = 540 degrees
Given angles: 120, 80, 100, 90
Equation: 120 + 80 + 100 + 90 + x = 540
Solution:
390 + x = 540
x = 150 degrees
Example 2: Finding an Exterior Angle
Given angles: 90, 90, 120, x
Equation: 90 + 90 + 120 + x = 360
Solution:
300 + x = 360
x = 60 degrees
Example 3: Regular Decagon
Definition: A decagon has 10 sides
Exterior angle calculation:
360 / 10 =
36 degrees
Interior angle relationships:
Interior angle + Exterior angle = 180 degrees
If exterior angle is 36 degrees, then interior angle = 180 - 36 = 144 degrees.
Example 4: Type of Polygon from Interior Angles
Given that sum of interior angles = 2,340 degrees
Use the interior angle formula:
2340 = (n - 2) * 180
Solve:
2340 / 180 = 13
n - 2 = 13
n = 15
Conclusion: It is a
15-gon
.
Conclusion
Importance of understanding interior and exterior angles in polygons for problem-solving.
Encouragement to subscribe for more educational content and future videos.
📄
Full transcript
Understanding Interior and Exterior Angles in Polygons
Sum of Interior Angles
Formula:
(n - 2) * 180
n
= number of sides (or angles) in the polygon
Explanation:
Example with a polygon having 5 sides (pentagon):
Can be divided into 3 triangles
Sum of angles in triangles = 3 * 180 = 540 degrees
Therefore, sum of interior angles for a polygon with n sides = (n - 2) * 180
Measure of One Interior Angle
Formula:
(n - 2) * 180 / n
Application:
Used for regular polygons (equal angles and equal side lengths)
Gives the measure of one interior angle by dividing total interior angle sum by number of angles.
Measure of One Exterior Angle
Formula:
360 / n
Application:
Used for regular polygons to find one exterior angle.
Sum of Exterior Angles
Always add up to
360 degrees
regardless of polygon type (regular or irregular).
Visualization: Formed angles on the outside of the polygon when extended add up to 360 degrees (like a circle).
Example Problems
Example 1: Finding an Interior Angle
Given a polygon with 5 sides:
Calculate sum of angles:
(5 - 2) * 180 = 540 degrees
Given angles: 120, 80, 100, 90
Equation: 120 + 80 + 100 + 90 + x = 540
Solution:
390 + x = 540
x = 150 degrees
Example 2: Finding an Exterior Angle
Given angles: 90, 90, 120, x
Equation: 90 + 90 + 120 + x = 360
Solution:
300 + x = 360
x = 60 degrees
Example 3: Regular Decagon
Definition: A decagon has 10 sides
Exterior angle calculation:
360 / 10 =
36 degrees
Interior angle relationships:
Interior angle + Exterior angle = 180 degrees
If exterior angle is 36 degrees, then interior angle = 180 - 36 = 144 degrees.
Example 4: Type of Polygon from Interior Angles
Given that sum of interior angles = 2,340 degrees
Use the interior angle formula:
2340 = (n - 2) * 180
Solve:
2340 / 180 = 13
n - 2 = 13
n = 15
Conclusion: It is a
15-gon
.
Conclusion
Importance of understanding interior and exterior angles in polygons for problem-solving.
Encouragement to subscribe for more educational content and future videos.