Find Measure Of Interior Angle Calculator

Calculate Interior Angle

Find the measure of each interior angle of a regular polygon by entering the number of sides.

What is the Measure of an Interior Angle of a Regular Polygon?

The measure of an interior angle of a regular polygon is the size of the angle formed inside the polygon at one of its vertices, between two adjacent sides. In a regular polygon, all interior angles are equal in measure. To find the measure of an interior angle, you first need to know the total sum of all interior angles, which depends on the number of sides the polygon has.

This calculator is useful for students, teachers, architects, and anyone working with geometric shapes who needs to quickly find the measure of an interior angle for a regular polygon.

A common misconception is that all polygons with the same number of sides have the same interior angle measures. This is only true for *regular* polygons, where all sides are equal in length and all interior angles are equal in measure.

Measure of Interior Angle Formula and Mathematical Explanation

The sum of the interior angles of any simple (non-self-intersecting) polygon with ‘n’ sides can be found using the formula:

Sum of Interior Angles = (n – 2) * 180°

Where ‘n’ is the number of sides of the polygon.

Since a regular polygon has all its interior angles equal, to find the measure of one interior angle, we divide the sum of the interior angles by the number of sides (or angles), ‘n’:

Measure of One Interior Angle = [(n – 2) * 180°] / n

Let’s break it down:

1. (n – 2): This represents the number of triangles you can divide the polygon into by drawing diagonals from one vertex.
2. (n – 2) * 180°: Since each triangle has an interior angle sum of 180°, this gives the total sum of interior angles of the polygon.
3. [(n – 2) * 180°] / n: Dividing the total sum by the number of angles (‘n’) gives the measure of each interior angle in a regular polygon.

Variables Table

Variable	Meaning	Unit	Typical Range
n	Number of sides of the regular polygon	None (integer)	n ≥ 3
Sum	Sum of all interior angles	Degrees (°)	≥ 180°
Angle	Measure of one interior angle	Degrees (°)	≥ 60° and < 180°

Practical Examples

Example 1: Regular Pentagon

A regular pentagon has 5 sides (n=5).

• Sum of interior angles = (5 – 2) * 180° = 3 * 180° = 540°
• Measure of one interior angle = 540° / 5 = 108°

So, each interior angle in a regular pentagon measures 108°.

Example 2: Regular Octagon

A regular octagon has 8 sides (n=8).

• Sum of interior angles = (8 – 2) * 180° = 6 * 180° = 1080°
• Measure of one interior angle = 1080° / 8 = 135°

So, each interior angle in a regular octagon measures 135°.

How to Use This Measure of Interior Angle Calculator

1. Enter the Number of Sides: Input the number of sides (n) of the regular polygon into the “Number of Sides (n)” field. Remember, a polygon must have at least 3 sides.
2. View the Results: The calculator will automatically update and display:
   • The measure of one interior angle (primary result).
   • The sum of all interior angles.
   • The formula used with your input value.
3. Reset: Click the “Reset” button to clear the input and results and return to the default value (5 sides).
4. Copy Results: Click “Copy Results” to copy the calculated values and formula to your clipboard.

This calculator helps you quickly determine the measure of interior angle without manual calculation.

Key Factors That Affect the Measure of an Interior Angle

1. Number of Sides (n): This is the primary factor. As the number of sides increases, the measure of each interior angle also increases, approaching 180° but never reaching it.
2. Regularity of the Polygon: The formula [(n – 2) * 180°] / n is only valid for regular polygons where all interior angles are equal. For irregular polygons, each interior angle can have a different measure, although their sum is still (n – 2) * 180°.
3. The Sum (n-2) * 180°: The total sum of interior angles directly influences the individual angle measure when divided by ‘n’. The more sides, the larger the sum.
4. Approaching 180°: As ‘n’ gets very large, the term (n-2)/n approaches 1, so the measure of the interior angle approaches 180°. A polygon with an infinite number of sides would resemble a circle, but a polygon is defined by straight sides.
5. Geometric Constraints: A polygon must be a closed figure with straight sides. The minimum number of sides is 3 (a triangle).
6. Exterior Angle Relationship: The measure of an exterior angle of a regular polygon is 360°/n. The interior and exterior angles at any vertex add up to 180°. So, Interior Angle = 180° – (360°/n), which is equivalent to [(n – 2) * 180°] / n.

Frequently Asked Questions (FAQ)

What is the minimum number of sides a polygon can have?

A polygon must have at least 3 sides, forming a triangle.

What is the minimum measure of an interior angle for a regular polygon?

For a regular triangle (equilateral), the interior angle is (3-2)*180/3 = 60°.

Can the measure of an interior angle of a regular polygon be 180°?

No, the measure of an interior angle of a simple regular polygon is always less than 180°. It approaches 180° as the number of sides increases infinitely, but never reaches it for a finite number of sides.

What is the formula for the sum of interior angles?

The sum of interior angles of any simple n-sided polygon is (n – 2) * 180°.

How do I find the measure of an interior angle if the polygon is irregular?

For an irregular polygon, the interior angles are not all equal. You would need more information about the specific angles or side lengths to determine each individual angle. The sum is still (n-2)*180°, but they are not evenly distributed.

What is a regular polygon?

A regular polygon is a polygon that is both equilateral (all sides are equal in length) and equiangular (all interior angles are equal in measure).

What happens to the interior angle as the number of sides increases?

As the number of sides ‘n’ increases, the measure of each interior angle of a regular polygon increases and gets closer and closer to 180°.

Can I use this calculator for star polygons?

No, this calculator and formula are for simple regular polygons (convex or non-convex but not self-intersecting in the way star polygons do). Star polygons have a different formula for their interior angles.

Related Tools and Internal Resources

• Area of Regular Polygon Calculator: Calculate the area of a regular polygon given the number of sides and side length or apothem.
• Exterior Angle Calculator: Find the measure of an exterior angle of a regular polygon.
• Triangle Angle Calculator: Calculate angles in various types of triangles.
• Basic Geometry Formulas: A guide to common geometric formulas.
• Properties of Quadrilaterals: Learn about different types of four-sided polygons.
• Circle Calculator: Calculate circumference, area, and other properties of a circle, which a polygon with infinite sides approaches.

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