Regular Polygon Calculator

What is a Regular Polygon?

A regular polygon is a closed, two-dimensional shape with straight sides, where all sides have equal length and all interior angles are equal. Examples include equilateral triangles, squares, regular pentagons, and so on.

How to Calculate Regular Polygon Properties

To fully understand a regular polygon, we need to calculate several key properties: the number of sides, side length, inradius (apothem), circumradius, interior and exterior angles, area, and perimeter. Each of these properties provides unique information about the polygon's size and shape.

Formulas

Here are the essential formulas for a regular polygon:

1. Inradius (r) or Apothem:

\[ r = \frac{a}{2\tan(\frac{\pi}{n})} \]

2. Circumradius (R):

\[ R = \frac{a}{2\sin(\frac{\pi}{n})} \]

3. Side length (a):

\[ a = 2r\tan(\frac{\pi}{n}) \]

4. Interior angle (x):

\[ x = \frac{(n-2)\pi}{n} \]

5. Exterior angle (y):

\[ y = \frac{2\pi}{n} \]

6. Area (A):

\[ A = \frac{1}{2}na^2\cot(\frac{\pi}{n}) \]

7. Perimeter (P):

\[ P = na \]

Where:

• \(n\) is the number of sides
• \(a\) is the side length
• \(r\) is the inradius (apothem)
• \(R\) is the circumradius
• \(\pi\) (pi) is approximately 3.14159

Calculation Steps

1. Determine the number of sides (n) and one other property (usually side length, inradius, or circumradius)
2. Calculate the remaining properties using the formulas above
3. Compute the interior and exterior angles
4. Calculate the area and perimeter

Example and Visual Representation

Let's calculate the properties of a regular pentagon (n = 5) with an inradius (apothem) of 5 units:

1. Number of sides: \(n = 5\)
2. Inradius: \(r = 5\) units
3. Side length: \(a = 2r\tan(\frac{\pi}{n}) = 2(5)\tan(\frac{\pi}{5}) \approx 5.88\) units
4. Circumradius: \(R = \frac{a}{2\sin(\frac{\pi}{n})} = \frac{5.88}{2\sin(\frac{\pi}{5})} \approx 6.18\) units
5. Interior angle: \(x = \frac{(n-2)\pi}{n} = \frac{3\pi}{5} \approx 108°\)
6. Exterior angle: \(y = \frac{2\pi}{n} = \frac{2\pi}{5} = 72°\)
7. Area: \(A = \frac{1}{2}na^2\cot(\frac{\pi}{n}) = \frac{1}{2}(5)(5.88)^2\cot(\frac{\pi}{5}) \approx 86.60\) square units
8. Perimeter: \(P = na = 5(5.88) = 29.40\) units

Here's a visual representation of this regular pentagon:

In this diagram, you can see the regular pentagon with its key properties labeled. The blue circle represents the inscribed circle (with radius r), and the pink dashed circle represents the circumscribed circle (with radius R). The green line shows a side length (a), and the orange line represents the apothem (r). The interior and exterior angles are also labeled.