Measuring the External Angle of a Regular Hexagon

Understanding the geometric properties of shapes is fundamental in both mathematics and practical applications. One key aspect is determining the measure of the external angles of polygons. In this article, we will explore how to calculate the external angle of a regular hexagon and extend this concept to other regular polygons.

Calculating the External Angle of a Regular Hexagon

A regular hexagon is a six-sided polygon with all sides and angles equal. To find the measure of an external angle of a regular hexagon, we can use the formula:

External Angle Formula

[ text{External angle} frac{360^circ}{n} ]

where ( n ) is the number of sides. For a regular hexagon, ( n 6 ).

Substituting the value of ( n ) into the formula, we get:

[ text{External angle} frac{360^circ}{6} 60^circ ]

Therefore, the measure of the external angle of a regular hexagon is ( 60^circ ).

The Sum of External Angles in Polygons

The sum of the exterior angles of any polygon is always ( 360^circ ). For regular polygons, all exterior angles are congruent, meaning they are equal in measure.

For a regular hexagon, the sum of the exterior angles is ( 360^circ ), and each exterior angle measures ( 60^circ ).

Understanding Interior and Exterior Angles

At each vertex of a regular hexagon, the sum of an interior angle and an exterior angle is ( 180^circ ). Since a regular hexagon has six sides, the total sum of the interior angles is calculated as:

[ 180^circ (n - 2) 180^circ (6 - 2) 720^circ ]

Each interior angle of a regular hexagon is:

[ frac{720^circ}{6} 120^circ ]

To find the external angle at each vertex, we subtract the interior angle from ( 180^circ ):

[ 180^circ - 120^circ 60^circ ]

This confirms that the external angle of a regular hexagon is indeed ( 60^circ ).

Generalizing to Other Regular Polygons

The same principle can be applied to other regular polygons. For instance:

Regular Heptagon

A regular heptagon is a seven-sided polygon with all sides and angles equal. The external angle for a regular heptagon is:

[ frac{360^circ}{7} 51.43^circ ]

Similarly, for any regular polygon with ( n ) sides, the external angle can be calculated using:

[ frac{360^circ}{n} ]

Summarizing the Key Concepts

The following table summarizes the key points:

Polygon Number of Sides (n) Sum of Interior Angles Each Interior Angle External Angle Hexagon 6 720° 120° 60° Heptagon 7 900° 128.57° 51.43°

Conclusion

Understanding the properties of external angles in regular polygons is essential for various mathematical and real-world applications. By using the formula and understanding the relationship between interior and exterior angles, we can easily calculate the external angle of a regular hexagon and generalize this to other regular polygons.

Frequently Asked Questions

Q1: How do I find the measure of the external angle of a regular hexagon?
A1: Use the formula [ frac{360^circ}{n}, ] where ( n 6 ). This gives an external angle of ( 60^circ ).

Q2: What is the sum of the exterior angles of a polygon?
A2: The sum of the exterior angles of any polygon is ( 360^circ ).

Q3: What is the interior angle of a regular hexagon?
A3: Each interior angle of a regular hexagon is ( 120^circ ).