Find Measure Of Exterior Angle Calculator

Calculate Exterior Angle

What is a Measure of Exterior Angle Calculator?

A measure of exterior angle calculator is a tool used to determine the size of an exterior angle of a polygon. For a regular polygon, all exterior angles are equal, and their measure can be found by dividing 360 degrees by the number of sides. For any convex polygon, an exterior angle at a given vertex is supplementary to the interior angle at that vertex (they add up to 180 degrees). This measure of exterior angle calculator can help students, teachers, and anyone working with geometry quickly find these angles.

You can use this measure of exterior angle calculator by either providing the number of sides (for a regular polygon) or the measure of one interior angle (for any polygon at that vertex). The sum of the exterior angles of any convex polygon is always 360 degrees.

Common misconceptions include thinking that the sum of exterior angles changes with the number of sides (it’s always 360° for convex polygons) or confusing interior and exterior angles.

Measure of Exterior Angle Formula and Mathematical Explanation

The formulas used by the measure of exterior angle calculator depend on the information you have:

1. For a Regular Polygon (given the number of sides, n):

   The sum of the exterior angles of any convex polygon is 360°. Since a regular polygon has equal exterior angles, the measure of one exterior angle is:

   Exterior Angle = 360° / n

   Where ‘n’ is the number of sides. The corresponding interior angle is:

   Interior Angle = 180° - (360° / n) or Interior Angle = (180° * (n-2)) / n

2. For Any Convex Polygon (given one interior angle, I):

   An interior angle and its adjacent exterior angle form a linear pair, meaning they add up to 180°.

   Exterior Angle = 180° - Interior Angle (I)

Variables Used

Variable	Meaning	Unit	Typical Range
n	Number of sides of the polygon	–	n ≥ 3
Exterior Angle	Measure of one exterior angle	Degrees (°)	0° < Exterior Angle < 180° (typically < 120° for n≥3)
Interior Angle (I)	Measure of one interior angle	Degrees (°)	0° < Interior Angle < 180° (typically > 60° for n≥3)
Sum of Exterior Angles	Total of all exterior angles	Degrees (°)	360° (for convex polygons)

Practical Examples (Real-World Use Cases)

Let’s see how the measure of exterior angle calculator works with some examples:

Example 1: Regular Pentagon

• Input: Regular Polygon, Number of Sides = 5
• Calculation (Exterior): 360° / 5 = 72°
• Calculation (Interior): 180° – 72° = 108°
• Output: Exterior Angle = 72°, Interior Angle = 108°, Sum of Exterior Angles = 360°

Example 2: A Polygon with a Known Interior Angle

• Input: Any Polygon, Interior Angle = 120°
• Calculation (Exterior): 180° – 120° = 60°
• Output: Exterior Angle at that vertex = 60°, Sum of Exterior Angles (if convex) = 360°

This measure of exterior angle calculator is useful in fields like architecture (designing structures with specific angles), surveying, and art.

How to Use This Measure of Exterior Angle Calculator

1. Select Calculation Method: Choose whether you know the “Number of Sides” (for a regular polygon) or “one Interior Angle”.
2. Enter Input Value:
   • If you selected “Number of Sides”, enter a whole number greater than or equal to 3.
   • If you selected “Interior Angle”, enter the angle in degrees (between 0 and 180).
3. View Results: The calculator will instantly display the measure of the exterior angle, the sum of exterior angles, and the corresponding interior angle (if applicable and calculable).
4. Interpret Results: The primary result is the measure of one exterior angle. Intermediate values provide context.
5. Use the Chart: The chart visually represents how the exterior angle changes with the number of sides for regular polygons.

Our measure of exterior angle calculator simplifies these calculations.

Key Factors That Affect Exterior Angle Results

• Number of Sides (for regular polygons): As the number of sides increases, the measure of each exterior angle decreases (since the sum is constant at 360°).
• Measure of the Interior Angle: The exterior angle is directly dependent on the interior angle at the same vertex; as the interior angle increases, the exterior angle decreases, and vice-versa.
• Type of Polygon (Regular vs. Irregular): For regular polygons, all exterior angles are equal. For irregular polygons, exterior angles can vary at each vertex, but their sum is still 360° (for convex ones). Our measure of exterior angle calculator handles both regular (with ‘n’) and irregular (with ‘I’) cases for one vertex.
• Convexity: The rule that the sum of exterior angles is 360° applies to convex polygons. For concave polygons, the definition and sum of exterior angles are different. This calculator assumes convex polygons.
• Accuracy of Input: Precise input of the number of sides or interior angle is crucial for an accurate output from the measure of exterior angle calculator.
• Units: Angles are measured in degrees. Ensure your input for the interior angle is in degrees.

For more complex shapes, you might need a geometry formulas guide.

Frequently Asked Questions (FAQ)

What is an exterior angle of a polygon?

An exterior angle is formed by one side of a polygon and the extension of an adjacent side, outside the polygon.

What is the sum of the exterior angles of any convex polygon?

The sum of the exterior angles of any convex polygon is always 360 degrees, regardless of the number of sides.

How do I find the exterior angle of a regular polygon?

Divide 360 by the number of sides (n). Our measure of exterior angle calculator does this automatically.

How do I find an exterior angle if I know the interior angle?

Subtract the interior angle from 180 degrees (Exterior = 180 – Interior). The measure of exterior angle calculator can also do this.

Can a polygon have an exterior angle greater than 180 degrees?

No, for convex polygons, exterior angles are always less than 180 degrees.

How many exterior angles does a polygon have?

A polygon has as many exterior angles as it has sides (and vertices), considering one exterior angle at each vertex.

Does this calculator work for concave polygons?

This measure of exterior angle calculator is primarily designed for convex polygons, where the sum of exterior angles is 360°. The concept is different for concave polygons.

What if I only know the exterior angle of a regular polygon and want to find the number of sides?

You can rearrange the formula: Number of Sides (n) = 360 / Exterior Angle. You could use our regular polygon calculator for more details.

Related Tools and Internal Resources

• Sum of Interior Angles Calculator: Calculate the total sum of interior angles for any polygon.
• Polygon Area Calculator: Find the area of various polygons.
• Triangle Angle Calculator: Focus specifically on the angles within a triangle.
• Quadrilateral Properties: Explore properties of four-sided figures.
• Geometry Formulas Explained: A guide to various geometry formulas and concepts.
• Regular Polygon Calculator: A tool focused on the properties of regular polygons.