Find the Measure of an Exterior Angle Calculator
This calculator helps you find the measure of an exterior angle of a polygon using different methods. Select the method and enter the required values.
Exterior Angle of Regular Polygons (3-10 sides)
Chart showing the exterior angle decreasing as the number of sides increases.
Exterior & Interior Angles of Regular Polygons
| Sides (n) | Interior Angle (°)/Side | Exterior Angle (°)/Side |
|---|---|---|
| 3 | 60.00 | 120.00 |
| 4 | 90.00 | 90.00 |
| 5 | 108.00 | 72.00 |
| 6 | 120.00 | 60.00 |
| 7 | 128.57 | 51.43 |
| 8 | 135.00 | 45.00 |
Table listing interior and exterior angles for regular polygons with 3 to 8 sides.
Understanding the Find the Measure of an Exterior Angle Calculator
The find the measure of an exterior angle calculator is a tool designed to quickly determine the measure of an exterior angle of a polygon. Whether you’re dealing with a regular polygon, know an interior angle, or have the remote interior angles of a triangle, this calculator provides the answer. An exterior angle is formed by one side of a polygon and the extension of an adjacent side.
What is an Exterior Angle of a Polygon?
An exterior angle of a polygon is the angle formed outside the polygon between one of its sides and the line extending from an adjacent side. At each vertex of a polygon, there are two exterior angles (which are vertically opposite and thus equal), but we usually consider just one. The sum of the exterior angles of any convex polygon, one at each vertex, is always 360 degrees.
This find the measure of an exterior angle calculator is useful for students, teachers, engineers, and anyone working with geometric shapes. Common misconceptions include thinking the sum varies with the number of sides (it’s always 360°) or confusing interior and exterior angles.
Exterior Angle Formulas and Mathematical Explanation
There are a few ways to calculate exterior angle measures depending on the information you have:
1. For a Regular Polygon
In a regular polygon (where all sides and angles are equal), the measure of each exterior angle is the same. Since the sum of all exterior angles is 360 degrees, and there are ‘n’ sides (and thus ‘n’ vertices), the formula is:
Exterior Angle = 360° / n
Where ‘n’ is the number of sides.
2. From an Interior Angle
An interior angle and its corresponding exterior angle form a linear pair, meaning they add up to 180 degrees.
Exterior Angle = 180° – Interior Angle
3. For a Triangle (using Remote Interior Angles)
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles (the interior angles that are not adjacent to the exterior angle).
Exterior Angle = Remote Interior Angle 1 + Remote Interior Angle 2
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| n | Number of sides of the polygon | None (integer) | ≥ 3 |
| Interior Angle | The angle inside the polygon at a vertex | Degrees (°) | 0° < Angle < 180° |
| Exterior Angle | The angle outside the polygon formed by one side and an extension of an adjacent side | Degrees (°) | 0° < Angle < 180° (usually) |
| Remote Interior Angle | Interior angles of a triangle not adjacent to the exterior angle | Degrees (°) | 0° < Angle < 180° |
Practical Examples (Real-World Use Cases)
Example 1: Regular Pentagon
You have a regular pentagon (5 sides). Using the find the measure of an exterior angle calculator in ‘Regular Polygon’ mode:
- Input: Number of Sides (n) = 5
- Calculation: Exterior Angle = 360° / 5 = 72°
- Output: Each exterior angle of a regular pentagon is 72°. Each interior angle is 180° – 72° = 108°.
Example 2: Triangle with Remote Interior Angles
A triangle has two remote interior angles measuring 50° and 70°. Using the find the measure of an exterior angle calculator in ‘Triangle’ mode:
- Input: Remote Interior Angle 1 = 50°, Remote Interior Angle 2 = 70°
- Calculation: Exterior Angle = 50° + 70° = 120°
- Output: The exterior angle corresponding to the third vertex is 120°. The third interior angle is 180° – (50° + 70°) = 60°, and 180° – 60° = 120°.
How to Use This Find the Measure of an Exterior Angle Calculator
- Select Mode: Choose whether you are calculating for a “Regular Polygon”, using an “From Interior Angle”, or for a “Triangle (from remote interior angles)”.
- Enter Values:
- For ‘Regular Polygon’, enter the number of sides (n).
- For ‘From Interior Angle’, enter the measure of the interior angle.
- For ‘Triangle’, enter the measures of the two remote interior angles.
- View Results: The calculator will instantly display the exterior angle in the “Results” section, along with the mode and inputs used. The formula applied will also be shown.
- Use Chart and Table: The chart and table provide additional context for regular polygons.
The results from the exterior angle calculator give you the measure in degrees. This is crucial for various geometric and construction applications.
Key Factors That Affect Exterior Angle Results
- Number of Sides (n): For regular polygons, as ‘n’ increases, the exterior angle decreases (360/n).
- Measure of Interior Angle: The larger the interior angle, the smaller the corresponding exterior angle (180 – Interior).
- Measures of Remote Interior Angles (Triangle): The sum directly gives the exterior angle; larger remote angles mean a larger exterior angle.
- Type of Polygon: The formulas differ for regular vs. irregular polygons and specifically for triangles.
- Convexity: These formulas apply to convex polygons. For concave polygons, the concept of exterior angles is more complex.
- Accuracy of Input: Ensure your input angles or number of sides are correct for an accurate exterior angle calculation.
Frequently Asked Questions (FAQ)
A: The sum of the exterior angles (one at each vertex) of any convex polygon is always 360 degrees, regardless of the number of sides.
A: An octagon has 8 sides. Use the formula 360/n = 360/8 = 45 degrees. Our find the measure of an exterior angle calculator can do this.
A: For a convex polygon, exterior angles (as typically defined) are less than 180 degrees. If you consider reflex angles formed by extending a side, you could get larger values, but the standard exterior angle is 180 – interior angle.
A: At any vertex, the interior angle and the exterior angle are supplementary, meaning they add up to 180 degrees.
A: If it’s not regular, you can find the exterior angle at a specific vertex if you know the interior angle at that vertex (Exterior = 180 – Interior). For a triangle, you can also use the remote interior angles. The exterior angle calculator supports these.
A: Yes, a triangle is a polygon with 3 sides. You can use the “Regular Polygon” mode if it’s equilateral, or the “Triangle” mode if you know two interior angles.
A: The minimum number of sides is 3 (a triangle).
A: The standard formulas and the calculator are designed for convex polygons. Concave polygons have at least one interior angle greater than 180 degrees, and the exterior angle definition can be different.
Related Tools and Internal Resources
- Interior Angle Calculator: Calculate the interior angles of polygons.
- Polygon Area Calculator: Find the area of various polygons.
- Triangle Area Calculator: Calculate the area of a triangle using different formulas.
- Geometry Calculators: A collection of calculators for various geometric problems.
- Angle Conversion Tool: Convert between degrees, radians, and other units.
- Sum of Interior Angles Calculator: Find the sum of interior angles based on the number of sides.