Calculator For Finding Interior Angle

Calculate Interior Angles

Enter the number of sides of a regular polygon to find its interior and exterior angles.

Common Regular Polygon Angles

Table of interior, exterior, and sum of interior angles for common regular polygons.

Sides (n)	Polygon Name	Interior Angle (°)	Exterior Angle (°)	Sum of Interior Angles (°)
3	Triangle	60	120	180
4	Quadrilateral	90	90	360
5	Pentagon	108	72	540
6	Hexagon	120	60	720
7	Heptagon	128.57	51.43	900
8	Octagon	135	45	1080
9	Nonagon	140	40	1260
10	Decagon	144	36	1440

What is an Interior Angle Calculator?

An interior angle calculator is a tool used to determine the measure of an interior angle of a regular polygon given the number of its sides. A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). This calculator also typically provides the sum of the interior angles and the measure of an exterior angle for the regular polygon.

Students of geometry, architects, designers, engineers, and anyone working with geometric shapes can benefit from using an interior angle calculator to quickly find angle measures without manual calculation.

A common misconception is that the formula used by the interior angle calculator for regular polygons can find each individual interior angle of an irregular polygon. While the formula for the sum of interior angles ((n-2) * 180°) applies to both regular and irregular polygons, the formula for a single interior angle ((n-2) * 180° / n) only gives the measure of each angle if the polygon is regular and all angles are equal.

Interior Angle Calculator Formula and Mathematical Explanation

The formulas used by the interior angle calculator for a regular polygon with ‘n’ sides are:

• Sum of the Interior Angles: Sum = (n – 2) × 180°
• Each Interior Angle: Angle = (n – 2) × 180° / n
• Each Exterior Angle: Angle = 360° / n

The formula for the sum of interior angles comes from the fact that any polygon can be divided into (n-2) triangles by drawing diagonals from one vertex, and the sum of angles in each triangle is 180°.

For a regular polygon, since all interior angles are equal, we divide the sum by the number of sides (n) to get the measure of one interior angle. The exterior angle is found because the sum of an interior and its corresponding exterior angle is 180°, and the sum of all exterior angles is 360°.

Variables Used:

Variables used in the interior angle calculations.

Variable	Meaning	Unit	Typical Range
n	Number of sides of the regular polygon	None (integer)	n ≥ 3
Interior Angle	Measure of one interior angle	Degrees (°)	60° to almost 180°
Sum of Interior Angles	Total measure of all interior angles	Degrees (°)	180° and up
Exterior Angle	Measure of one exterior angle	Degrees (°)	Almost 180° down to almost 0°

Practical Examples (Real-World Use Cases)

Let’s see how the interior angle calculator works with a couple of examples:

Example 1: Regular Pentagon

Suppose you are designing a logo based on a regular pentagon and need to know its angles.

• Input: Number of sides (n) = 5
• Sum of Interior Angles = (5 – 2) × 180° = 3 × 180° = 540°
• Each Interior Angle = 540° / 5 = 108°
• Each Exterior Angle = 360° / 5 = 72°

The interior angle calculator quickly shows each interior angle of the regular pentagon is 108°.

Example 2: Regular Octagon

Imagine you’re tiling a floor with regular octagonal tiles and need to fit them together.

• Input: Number of sides (n) = 8
• Sum of Interior Angles = (8 – 2) × 180° = 6 × 180° = 1080°
• Each Interior Angle = 1080° / 8 = 135°
• Each Exterior Angle = 360° / 8 = 45°

The interior angle calculator tells you each interior angle is 135°, which helps in understanding how the tiles will meet or what shape of tile is needed to fill the gaps.

How to Use This 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. The minimum value is 3.
2. Calculate: The calculator automatically updates as you type, or you can click “Calculate”.
3. View Results:
   • The “Primary Result” shows the measure of each interior angle.
   • “Intermediate Results” display the polygon’s name (if common), the sum of its interior angles, and the measure of each exterior angle.
   • The formula used is also shown.
4. Reset: Click “Reset” to return the input to the default value (5).
5. Copy Results: Click “Copy Results” to copy the main results and input to your clipboard.

Use the results from the interior angle calculator to aid in geometry problems, design projects, or any application requiring knowledge of regular polygon angles.

Key Factors That Affect Interior Angle Calculator Results

For a regular polygon, the results of the interior angle calculator depend solely on one factor:

1. Number of Sides (n): This is the fundamental determinant. As ‘n’ increases, the sum of interior angles increases, and each individual interior angle also increases, approaching 180°. The exterior angle decreases, approaching 0°.
2. Regularity of the Polygon: The formulas for each interior and exterior angle provided by this interior angle calculator are specifically for regular polygons, where all sides and angles are equal. If the polygon is irregular, only the sum of interior angles ((n-2)*180°) can be determined by ‘n’ alone; individual angles would require more information.
3. Convexity: The formulas assume a convex polygon (no interior angle greater than 180°). For star polygons or other non-convex shapes, the definition and calculation of “interior” angles become more complex. This interior angle calculator is for convex regular polygons.
4. Units: The calculator provides results in degrees. If you need radians, you would convert (180° = π radians).
5. Accuracy of ‘n’: ‘n’ must be an integer greater than or equal to 3.
6. Dimensionality: These calculations are for 2-dimensional polygons.

Frequently Asked Questions (FAQ)

What is a regular polygon?

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

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

A polygon must have at least 3 sides (a triangle).

How do you find the sum of the interior angles of any polygon?

The sum of the interior angles of any convex polygon (regular or irregular) with ‘n’ sides is given by the formula (n – 2) × 180°.

Can I use this calculator for an irregular polygon?

You can use the formula (n-2)*180° to find the sum of interior angles for an irregular polygon, but the formula (n-2)*180°/n for each angle only works if it’s regular. Our interior angle calculator is designed for regular polygons for individual angle calculation.

What is an exterior angle?

An exterior angle of a regular polygon is the angle formed between one side of the polygon and the extension of an adjacent side. Its measure is 360°/n.

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

As the number of sides ‘n’ increases, the interior angle of a regular polygon gets larger and approaches 180°. The polygon looks more and more like a circle.

Are the angles always in degrees?

Yes, this interior angle calculator provides results in degrees.

How does the exterior angle relate to the interior angle?

For any polygon, an interior angle and its corresponding exterior angle add up to 180°.