Polygon Calculator

- May 11, 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 visualisations 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

Understanding the Polygon Calculator

The Polygon Calculator is a handy tool designed to help you calculate various properties of regular polygons. You can find out the area, perimeter, interior angles, and much more, all in one place. This calculator offers multiple methods for calculations, making it very user-friendly. Whether you're a student learning about shapes or a professional needing quick calculations, this tool is perfect for you.

Selecting Your Calculation Method

One of the great features of the Polygon Calculator is its flexibility in calculation methods. You can choose from several options based on the information you have available. Here are the primary methods you can select:

  • Number of Sides and Side Length
  • Number of Sides and Perimeter
  • Number of Sides and Area
  • Number of Sides and Inradius
  • Number of Sides and Circumradius

Inputting Your Values

After selecting your preferred calculation method, you’ll need to input your values. For example, if you choose to calculate based on the number of sides and side length, just enter the number of sides and the length of one side. The calculator will take care of the rest, providing accurate results for all the necessary properties of your polygon.

Getting Accurate Results with the Polygon Calculator

The Polygon Calculator delivers precise results quickly. You will receive detailed outputs such as the side length, perimeter, area, inradius, and circumradius. Additionally, it shows the interior and exterior angles along with the sum of the interior angles. This comprehensive information is valuable for anyone studying Geometry or working on practical applications.

Visualisation of Your Polygon

A unique aspect of this calculator is its ability to provide visualisations of your polygon based on the input values. You can see a graphical representation that helps you understand the shape better. The visualisation is accompanied by a legend and labels to guide you, making your learning experience more interactive and engaging.

Calculating with Steps and Formulas

The calculator not only gives you results but also outlines the calculation steps. This feature is particularly helpful for students who want to learn how to arrive at the answers independently. Additionally, it includes essential formulas used for calculations, ensuring you understand the underlying maths involved.

Key Properties of Regular Polygons

Understanding regular polygons is essential in geometry. Here are some key properties:

  • All sides are equal in length.
  • All interior angles are equal.
  • The sum of the interior angles can be calculated using the formula: (n-2) × 180°.
  • Each exterior angle can be found using: 360°/n.

Applications of the Polygon Calculator

The Polygon Calculator is not only useful for academic purposes but also has practical applications. Architects, designers, and engineers use shapes in their work. Regular polygons appear in everyday items and structures, such as:

  • Traffic signs, like stop signs that are octagonal.
  • Patterns in design and logos.
  • Structures in architecture that rely on polygonal shapes for stability.