Polygon Calculator

How to Calculate Polygon Properties

A polygon is a closed shape with straight sides. Regular polygons have all sides of equal length and all interior angles equal. Understanding how to calculate various properties of a polygon is crucial in geometry, architecture, and many real-world applications. Here's a comprehensive guide on how to perform these calculations:

Polygon Formulas

The key formulas for calculating polygon properties are:

• Perimeter (P) = \(ns\)
• Area (A) = \(\frac{ns^2}{4\tan(\frac{\pi}{n})}\)
• Apothem (a) = \(\frac{s}{2\tan(\frac{\pi}{n})}\)
• Circumradius (R) = \(\frac{s}{2\sin(\frac{\pi}{n})}\)
• Inradius (r) = Apothem = \(\frac{s}{2\tan(\frac{\pi}{n})}\)
• Interior Angle = \(\frac{(n-2) \times 180°}{n}\)
• Exterior Angle = \(\frac{360°}{n}\)

Where \(n\) is the number of sides and \(s\) is the length of one side of the polygon.

Calculation Steps

1. Identify the number of sides of the polygon.
2. Determine the given property of the polygon (side length, perimeter, area, or apothem).
3. If the side length is not given directly, calculate it using the appropriate formula.
4. Once the side length is known, use the formulas to calculate all other properties.
5. Round the results to an appropriate number of decimal places if necessary.

Example Calculation

Let's calculate the properties of a regular pentagon (5 sides) with a side length of 10 units:

1. Given: \(n = 5\), \(s = 10\) units
2. Perimeter: \(P = ns = 5 \times 10 = 50\) units
3. Area: \(A = \frac{ns^2}{4\tan(\frac{\pi}{n})} = \frac{5 \times 10^2}{4\tan(\frac{\pi}{5})} \approx 172.05\) square units
4. Apothem: \(a = \frac{s}{2\tan(\frac{\pi}{n})} = \frac{10}{2\tan(\frac{\pi}{5})} \approx 6.88\) units
5. Circumradius: \(R = \frac{s}{2\sin(\frac{\pi}{n})} = \frac{10}{2\sin(\frac{\pi}{5})} \approx 8.51\) units
6. Inradius: \(r = \text{Apothem} \approx 6.88\) units
7. Interior Angle: \(\frac{(n-2) \times 180°}{n} = \frac{(5-2) \times 180°}{5} = 108°\)
8. Exterior Angle: \(\frac{360°}{n} = \frac{360°}{5} = 72°\)

Visual Representation

This diagram illustrates a regular pentagon with its side (\(s\)), circumradius (\(R\)), and apothem (\(a\)) labeled. The pentagon is drawn in blue, the radius in red, and the apothem in green.