Polygon Calculator

Category: Geometry
- March 21, 2025
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Calculate various properties of regular polygons including area, perimeter, interior angles, and more. This calculator supports multiple calculation methods and provides visualizations with step-by-step explanations.

Select Calculation Method

Advanced Options

About Regular Polygons in Geometry

A regular polygon is a polygon that is equiangular (all angles are equal) and equilateral (all sides have the same length). Regular polygons are important in geometry, architecture, design, and many other fields.

Key Properties and Formulas
  • Number of Sides (n): The total count of sides and vertices in the polygon.
  • Side Length (s): The length of each side of the polygon.
  • Perimeter (P): P = n × s (The sum of all side lengths)
  • Interior Angle (θint): θint = (n − 2) × 180° / n (The angle inside the polygon at each vertex)
  • Exterior Angle (θext): θext = 360° / n (The angle formed by extending a side)
  • Inradius (r): r = s / (2 × tan(π/n)) (The radius of the inscribed circle)
  • Circumradius (R): R = s / (2 × sin(π/n)) (The radius of the circumscribed circle)
  • Area (A): A = (n × s²) / (4 × tan(π/n))
Common Regular Polygons
  • Triangle (n=3): Also known as an equilateral triangle.
  • Square (n=4): Has four equal sides and angles of 90°.
  • Pentagon (n=5): Has five equal sides and interior angles of 108°.
  • Hexagon (n=6): Has six equal sides and interior angles of 120°.
  • Octagon (n=8): Has eight equal sides and interior angles of 135°.
  • Decagon (n=10): Has ten equal sides and interior angles of 144°.
  • Dodecagon (n=12): Has twelve equal sides and interior angles of 150°.
Applications

Regular polygons have many practical applications:

  • Architecture: Building layouts, decorative elements, floor plans
  • Design: Logos, patterns, product shapes
  • Engineering: Mechanical parts, gear design, structural elements
  • Urban Planning: Traffic roundabouts, public spaces, park layouts
  • Nature: Many natural structures approximate regular polygons (honeycomb cells, crystal structures)
  • Traffic Signs: Stop signs are regular octagons

What Is the Polygon Calculator?

The Polygon Calculator is an interactive tool that helps you calculate essential properties of regular polygons. Whether you're a student, educator, designer, or engineer, this tool allows you to quickly determine values such as area, perimeter, angles, inradius, and circumradius for regular polygons based on different known inputs.

Key Features

  • Supports multiple input types (side length, perimeter, area, inradius, circumradius)
  • Automatically calculates all related polygon properties
  • Visualizes the polygon and its dimensions on a diagram
  • Provides step-by-step calculations for better understanding
  • Offers unit selection and angle format (degrees or radians)
  • Supports common units: mm, cm, m, in, ft

Formulas Used

Area of a Regular Polygon:
\( A = \frac{n \times s^2}{4 \times \tan\left(\frac{\pi}{n}\right)} \)
Perimeter:
\( P = n \times s \)
Interior Angle:
\( \theta_{\text{int}} = \frac{(n - 2) \times 180^\circ}{n} \)
Exterior Angle:
\( \theta_{\text{ext}} = \frac{360^\circ}{n} \)
Inradius:
\( r = \frac{s}{2 \times \tan\left(\frac{\pi}{n}\right)} \)
Circumradius:
\( R = \frac{s}{2 \times \sin\left(\frac{\pi}{n}\right)} \)

How to Use the Calculator

  1. Select your calculation method. Choose from options like "Number of Sides and Side Length" or "Number of Sides and Area".
  2. Enter the known values into the fields provided. You can also choose your preferred units.
  3. Adjust settings like decimal places, angle units, and whether to display the visualization or step-by-step process.
  4. Click the Calculate button to get your results.
  5. Review the output, which includes all relevant polygon properties, a labeled diagram, and calculation steps.

Who Can Benefit From This Tool?

  • Students and Teachers: Easily explore and demonstrate polygon properties with visual aids.
  • Designers: Create precise patterns and geometric shapes for artwork, logos, or layouts.
  • Engineers and Architects: Calculate structural dimensions or components with regular polygonal geometry.
  • Anyone Curious: Quickly calculate values for regular polygons without doing manual math.

Frequently Asked Questions (FAQ)

What is a regular polygon?

A regular polygon is a shape with all sides and all angles equal. Common examples include equilateral triangles, squares, and regular hexagons.

What is the minimum number of sides I can enter?

You must enter at least 3 sides, as that’s the smallest possible polygon (a triangle).

Can I switch between degrees and radians?

Yes. Use the dropdown under "Advanced Options" to choose your preferred angle unit.

Why is the visualization not showing?

Make sure the "Show Visualization" checkbox is selected. Also, check that your input values are valid.

What happens if I enter invalid values?

The calculator will prompt you with an alert message to correct the input. Only positive numbers and valid side counts are accepted.

Why Use This Calculator?

This tool saves time and reduces errors when working with regular polygons. It offers a clear visual representation, supports unit conversions, and walks you through the formulas step-by-step. Whether you're checking homework, designing graphics, or planning a build, it gives you quick and accurate results without needing to remember or manually apply complex formulas.