Understanding Exterior Angles of a Polygon

Oct 13, 2025 by miodeya.com 93 views

Understanding Exterior Angles of a Polygon

Exterior angles of a polygon are formed by extending the sides of the polygon. The sum of the exterior angles of any polygon is always 360°. This property is a fundamental concept in geometry and is used to find missing exterior angles.

The Sum of Exterior Angles of a Polygon

The sum of the exterior angles of a polygon is a constant value, which is 360°. This is a universal property that applies to all polygons, regardless of the number of sides. The sum of the exterior angles of a polygon can be calculated using the formula:

Sum of exterior angles = 360°

Finding the Measure of the Last Exterior Angle

Given that the hexagon has exterior angles of 17°, 43°, 59°, 67°, and 127°, we can find the measure of the last exterior angle by subtracting the sum of the given exterior angles from 360°.

Calculating the Measure of the Last Exterior Angle

Let's calculate the measure of the last exterior angle:

Sum of given exterior angles = 17° + 43° + 59° + 67° + 127° = 313°

Now, subtract the sum of the given exterior angles from 360°:

Measure of the last exterior angle = 360° - 313° = 47°

Conclusion

The measure of the last exterior angle of the hexagon is 47°.

Exterior Angles of a Polygon: Key Takeaways

The sum of the exterior angles of any polygon is 360°.
The sum of the exterior angles of a polygon can be calculated using the formula: Sum of exterior angles = 360°.
To find the measure of a missing exterior angle, subtract the sum of the given exterior angles from 360°.

Real-World Applications of Exterior Angles

Exterior angles have numerous real-world applications, including:

Architecture: Exterior angles are used to design buildings and structures.
Engineering: Exterior angles are used to calculate the stress and strain on structures.
Navigation: Exterior angles are used to navigate and orient oneself in the world.

Practice Problems: Exterior Angles of a Polygon

What is the sum of the exterior angles of a pentagon?
Find the measure of the last exterior angle of a pentagon with exterior angles of 30°, 60°, 90°, and 120°.
What is the sum of the exterior angles of a triangle?

Solutions to Practice Problems

The sum of the exterior angles of a pentagon is 360°.
Let's calculate the measure of the last exterior angle:

Sum of given exterior angles = 30° + 60° + 90° + 120° = 300°

Now, subtract the sum of the given exterior angles from 360°:

Measure of the last exterior angle = 360° - 300° = 60°

The sum of the exterior angles of a triangle is 360°.

Conclusion

Exterior angles are a fundamental concept in geometry and have numerous real-world applications. The sum of the exterior angles of any polygon is 360°, and the measure of a missing exterior angle can be found by subtracting the sum of the given exterior angles from 360°.

Frequently Asked Questions

Exterior angles of a polygon are a fundamental concept in geometry, and many students and professionals have questions about them. Here are some frequently asked questions and answers about exterior angles of a polygon.

Q: What is the sum of the exterior angles of a polygon?

A: The sum of the exterior angles of any polygon is 360°.

Q: How do I find the measure of a missing exterior angle?

A: To find the measure of a missing exterior angle, subtract the sum of the given exterior angles from 360°.

Q: What is the formula for the sum of the exterior angles of a polygon?

A: The formula for the sum of the exterior angles of a polygon is:

Sum of exterior angles = 360°

Q: Can I use the formula to find the measure of a missing exterior angle?

A: Yes, you can use the formula to find the measure of a missing exterior angle. Simply subtract the sum of the given exterior angles from 360°.

Q: What is the relationship between the number of sides of a polygon and the sum of its exterior angles?

A: The sum of the exterior angles of a polygon is always 360°, regardless of the number of sides.

Q: Can I use exterior angles to find the measure of an interior angle of a polygon?

A: Yes, you can use exterior angles to find the measure of an interior angle of a polygon. The sum of an interior angle and its corresponding exterior angle is always 180°.

Q: What is the difference between an exterior angle and an interior angle of a polygon?

A: An exterior angle of a polygon is formed by extending a side of the polygon, while an interior angle is formed by two adjacent sides of the polygon.

Q: Can I use exterior angles to find the perimeter of a polygon?

A: Yes, you can use exterior angles to find the perimeter of a polygon. The perimeter of a polygon is equal to the sum of the lengths of its sides, which can be found using the exterior angles.

Q: What is the relationship between the sum of the exterior angles of a polygon and the number of its sides?

A: The sum of the exterior angles of a polygon is always 360°, regardless of the number of its sides.

Q: Can I use exterior angles to find the area of a polygon?

A: Yes, you can use exterior angles to find the area of a polygon. The area of a polygon can be found using the formula:

Area = (n * s^2) / (4 * tan(π/n))

where n is the number of sides and s is the length of a side.

Q: What is the relationship between the exterior angles of a polygon and its symmetry?

A: The exterior angles of a polygon are related to its symmetry. A polygon with equal exterior angles is symmetrical.

Q: Can I use exterior angles to find the measure of an angle in a circle?

A: Yes, you can use exterior angles to find the measure of an angle in a circle. The measure of an angle in a circle is equal to the sum of the exterior angles of the polygon formed by the arc.

Q: What is the relationship between the exterior angles of a polygon and its diagonals?

A: The exterior angles of a polygon are related to its diagonals. A diagonal of a polygon is a line segment that connects two non-adjacent vertices.

Q: Can I use exterior angles to find the measure of an angle in a triangle?

A: Yes, you can use exterior angles to find the measure of an angle in a triangle. The measure of an angle in a triangle is equal to the sum of the exterior angles of the polygon formed by the sides.

Q: What is the relationship between the exterior angles of a polygon and its circumcenter?

A: The exterior angles of a polygon are related to its circumcenter. The circumcenter of a polygon is the point where the perpendicular bisectors of the sides intersect.

Q: Can I use exterior angles to find the measure of an angle in a regular polygon?

A: Yes, you can use exterior angles to find the measure of an angle in a regular polygon. The measure of an angle in a regular polygon is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its incenter?

A: The exterior angles of a polygon are related to its incenter. The incenter of a polygon is the point where the angle bisectors intersect.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of sides?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of sides. The measure of an angle in a polygon with a given number of sides is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its centroid?

A: The exterior angles of a polygon are related to its centroid. The centroid of a polygon is the point where the medians intersect.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given perimeter?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given perimeter. The measure of an angle in a polygon with a given perimeter is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its area?

A: The exterior angles of a polygon are related to its area. The area of a polygon can be found using the formula:

Area = (n * s^2) / (4 * tan(π/n))

where n is the number of sides and s is the length of a side.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of diagonals?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of diagonals. The measure of an angle in a polygon with a given number of diagonals is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its circumradius?

A: The exterior angles of a polygon are related to its circumradius. The circumradius of a polygon is the distance from the circumcenter to any vertex.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given perimeter?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given perimeter. The measure of an angle in a polygon with a given number of sides and a given perimeter is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its inradius?

A: The exterior angles of a polygon are related to its inradius. The inradius of a polygon is the distance from the incenter to any vertex.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given area?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given area. The measure of an angle in a polygon with a given number of sides and a given area is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its centroidal mean?

A: The exterior angles of a polygon are related to its centroidal mean. The centroidal mean of a polygon is the average distance from the centroid to any vertex.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given centroidal mean?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given centroidal mean. The measure of an angle in a polygon with a given number of sides and a given centroidal mean is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its incenteral mean?

A: The exterior angles of a polygon are related to its incenteral mean. The incenteral mean of a polygon is the average distance from the incenter to any vertex.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given incenteral mean?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given incenteral mean. The measure of an angle in a polygon with a given number of sides and a given incenteral mean is equal to the sum of the exterior angles of the polygon.

Q: What is the relationship between the exterior angles of a polygon and its circumcenteral mean?

A: The exterior angles of a polygon are related to its circumcenteral mean. The circumcenteral mean of a polygon is the average distance from the circumcenter to any vertex.

Q: Can I use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given circumcenteral mean?

A: Yes, you can use exterior angles to find the measure of an angle in a polygon with a given number of sides and a given circumcenteral mean. The measure of an angle in a polygon with a given number of sides and a given circumcenteral mean is equal to the sum of