Find The Number Of Sides Of A Regular Polygon Calculator

Common Regular Polygons and Their Angles
Sides (n)	Name	Interior Angle (°)	Exterior Angle (°)	Sum of Interior Angles (°)
3	Triangle	60	120	180
4	Square	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

What is a Number of Sides of a Regular Polygon Calculator?

A Number of Sides of a Regular Polygon Calculator is a tool used to determine the number of sides (n) of a regular polygon when either its interior angle or exterior angle is known. 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 is particularly useful in geometry and related fields where the properties of regular polygons are studied.

Anyone studying geometry, from students to teachers and professionals like architects or designers, might use this calculator to quickly find the number of sides without manual calculation, or to verify their results. Common misconceptions include thinking any polygon with equal sides is regular (it must also have equal angles), or that the formulas apply to irregular polygons (they do not).

Number of Sides of a Regular Polygon Formula and Mathematical Explanation

The number of sides of a regular polygon can be found using the measure of one of its interior angles or one of its exterior angles.

1. Using the Interior Angle (a):

The formula for the measure of one interior angle (a) of a regular n-sided polygon is:

a = (n – 2) * 180 / n

To find ‘n’ when ‘a’ is known, we rearrange this formula:

an = 180n – 360
360 = 180n – an
360 = n(180 – a)
n = 360 / (180 – a)

2. Using the Exterior Angle (e):

The sum of the exterior angles of any convex polygon is 360 degrees. For a regular n-sided polygon, all exterior angles are equal, so each exterior angle (e) is:

e = 360 / n

Rearranging to find ‘n’ when ‘e’ is known:

n = 360 / e

Also, note that the interior angle and exterior angle at any vertex are supplementary: a + e = 180 degrees.

Variables Table

Variable	Meaning	Unit	Typical Range (for n>=3)
n	Number of sides	None (integer)	n ≥ 3
a	Interior angle	Degrees	60° ≤ a < 180°
e	Exterior angle	Degrees	0° < e ≤ 120°
S	Sum of interior angles	Degrees	S ≥ 180°

Our Number of Sides of a Regular Polygon Calculator uses these formulas based on your input.

Practical Examples (Real-World Use Cases)

Let’s see how the Number of Sides of a Regular Polygon Calculator works with some examples.

Example 1: Known Interior Angle

Suppose you are looking at a floor tile that is a regular polygon, and you measure one of its interior angles to be 140 degrees. You want to find out how many sides the tile has.

Input: Interior Angle = 140°
Using the formula n = 360 / (180 – a): n = 360 / (180 – 140) = 360 / 40 = 9
Result: The polygon has 9 sides (it’s a nonagon). The calculator would also show the exterior angle as 180 – 140 = 40°, and the sum of interior angles as (9-2)*180 = 1260°.

Example 2: Known Exterior Angle

Imagine you are designing a logo with a regular polygon and you decide its exterior angle should be 72 degrees for aesthetic reasons.

Input: Exterior Angle = 72°
Using the formula n = 360 / e: n = 360 / 72 = 5
Result: The polygon has 5 sides (it’s a pentagon). The calculator would also show the interior angle as 180 – 72 = 108°, and the sum of interior angles as (5-2)*180 = 540°.

Using the Number of Sides of a Regular Polygon Calculator makes these calculations quick and error-free.

How to Use This Number of Sides of a Regular Polygon Calculator

Using our Number of Sides of a Regular Polygon Calculator is simple:

Select Angle Type: Choose whether you know the ‘Interior Angle’ or the ‘Exterior Angle’ by selecting the corresponding radio button.
Enter Angle Value: Input the value of the known angle in degrees into the “Angle Value” field.
View Results: The calculator will automatically display the number of sides (n) as the primary result. It will also show the other angle (interior or exterior), the sum of the interior angles, and the formula used.
Reset: You can click the “Reset” button to clear the input and results and start over with default values.
Copy Results: Click “Copy Results” to copy the main result and intermediate values to your clipboard.

The calculator provides instant results, helping you understand the properties of the regular polygon based on the given angle. Ensure your angle input is valid for a regular polygon (e.g., interior angle must be less than 180 and correspond to n>=3).

Key Factors That Affect Number of Sides of a Regular Polygon Results

The primary factor determining the number of sides is the angle you input. Here’s a breakdown:

Angle Value: The precise value of the interior or exterior angle directly determines ‘n’. Small changes in the angle can lead to a different number of sides, or indicate that no regular polygon exists for that angle.
Angle Type (Interior/Exterior): Whether you input the interior or exterior angle changes the formula used (n = 360 / (180 – a) vs n = 360 / e), but they are related (a + e = 180), so a valid interior angle corresponds to a valid exterior angle and vice-versa.
Validity of the Angle: Not all angle values will result in a valid regular polygon (i.e., an integer n ≥ 3). The interior angle must be between 60° (for n=3) and 180° (as n approaches infinity), and the exterior angle between 0° and 120°. The calculator should indicate if the angle doesn’t correspond to a regular polygon with an integer number of sides.
Regularity Assumption: The formulas and the calculator assume the polygon is *regular* (all sides and angles are equal). If the polygon is irregular, these formulas do not apply for finding ‘n’ from a single angle.
Sum of Interior Angles: While not a direct input, the sum of interior angles (S = (n-2)*180) is directly tied to ‘n’. Knowing S allows you to find ‘n’ and thus the individual angles.
Units: The calculator assumes the input angle is in degrees. If your angle is in radians, you must convert it to degrees first.

Our Number of Sides of a Regular Polygon Calculator relies on these geometric principles.

Frequently Asked Questions (FAQ)

1. What is a regular polygon?: A regular polygon is a polygon that is both equilateral (all sides have the same length) and equiangular (all interior angles have the same measure).
2. Can I use this calculator for irregular polygons?: No, this Number of Sides of a Regular Polygon Calculator and the formulas used are specifically for regular polygons where all interior/exterior angles are equal.
3. What if the calculated number of sides is not an integer?: If the formulas yield a non-integer value for ‘n’, it means that no regular polygon exists with the given interior or exterior angle. For example, an interior angle of 100 degrees does not correspond to a regular polygon.
4. What is the minimum number of sides a polygon can have?: A polygon must have at least 3 sides (a triangle).
5. How are the interior and exterior angles related?: At any vertex of a convex polygon, the interior angle and the exterior angle are supplementary, meaning they add up to 180 degrees (a + e = 180°).
6. Why is the exterior angle formula simpler?: The sum of the exterior angles of any convex polygon is always 360 degrees. For a regular polygon with ‘n’ sides, all ‘n’ exterior angles are equal, so each is 360/n, leading to a direct formula for ‘n’.
7. What’s the largest possible interior angle for a regular polygon?: As the number of sides ‘n’ increases, the interior angle approaches 180 degrees, but never quite reaches it (e.g., for n=360, interior angle is 179°). A circle can be thought of as the limit as n approaches infinity.
8. Can I enter the sum of interior angles?: This specific calculator takes an individual interior or exterior angle. However, if you know the sum (S), you can find n using S = (n-2)*180 => n = S/180 + 2, and then find the individual angles.

Related Tools and Internal Resources

For more geometry and math calculations, explore these tools:

Polygon Area Calculator: Calculate the area of various polygons.
Triangle Angle Calculator: Find missing angles in a triangle.
Quadrilateral Properties: Learn about different quadrilaterals.
Circle Calculator: Calculate circumference, area, and other circle properties.
Geometry Formulas: A collection of common geometry formulas, including those for regular polygon properties.
Math Calculators: A directory of various math and geometry calculators, including the polygon side calculator.