Polygon Properties Calculator

Number of Sides: Side Length:

Looking to calculate the area of a regular polygon or find angles, perimeter, or radius of a polygon? The Polygon Properties Calculator at hcalculator is your go-to geometry polygon tool. Whether you’re working with a triangle, decagon, or 1000-gon, this calculator simplifies complex math with just a few inputs.

What Is a Polygon?

A polygon is a flat, 2D shape made of straight line segments that form a closed figure. Each line segment is a side, and the points where they meet are vertices. Common polygons include:

Names of Polygon Shapes by Number of Sides

Triangle – 3 sides
Quadrilateral – 4 sides
Pentagon – 5 sides
Hexagon—6 sides
Decagon – 10 sides
n-gon – n sides (as many as you want!)

Types of Polygons

Convex Polygons

All interior angles are less than 180°, and no vertices point inward.

Concave Polygons

At least one interior angle is more than 180°, with some vertices “caving in.”

Convex vs Concave: How to Tell

Draw diagonals—if all lie inside the polygon, it’s convex. If any lie outside, it’s concave.

Properties of a Polygon

Number of Sides (n)

The total number of edges or line segments.

Side Length (a)

The length of one side of the polygon.

Interior and Exterior Angles

Important for geometry and symmetry.

Inradius and Circumradius

Inradius (r): Distance from center to a side (incircle)
Circumradius (R): Distance from center to a vertex (circumcircle)

Apothem

A line from the center to the midpoint of a side—used for area calculation.

Area and Perimeter

Essential measurements in design, construction, and geometry.

Polygon Formulas

Here are the most important formulas used by the Polygon Calculator:

Area of a Regular Polygon

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Area = (n × a²) / (4 × tan (π / n))

Perimeter Formula

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Perimeter = n × a

Interior and Exterior Angles

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Interior Angle = [(n – 2) × 180°] / n Exterior Angle = 360° / n

Apothem Formula

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Apothem = a / (2 × tan (π / n))

Inradius Formula

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Inradius = a / (2 × tan (π / n))

Circumradius Formula

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Circumradius = a / (2 × sin (π / n))

How to Use the Polygon Calculator?

The Regular Polygon Calculator Tool on hcalculator simplifies all these calculations.

Inputs Required

Number of sides (n)
A known measurement: side length, inradius, or circumradius

Outputs You’ll Get

Area
Perimeter
Interior and exterior angles
Apothem
Inradius and circumradius

Example: Regular Hexagon Calculation

For a hexagon (n = 6) with side length a = 6:

Area: ≈ 93.53 units²
Perimeter: 36 units
Interior Angle: 120°
Inradius: ≈ 5.2
Circumradius: ≈ 6

Calculate the Properties of a Polygon

This polygon calculator helps with:

Area of a Polygon

Use:

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Area = (n × a²) / (4 × tan (π / n))

Perimeter

For regular polygons:

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Perimeter = n × a

For irregular: Add all sides manually.

Interior Angle

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Interior Angle = [(n – 2) × 180°] / n

Exterior Angle

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Exterior Angle = 360° / n

Inradius

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Inradius = a / (2 × tan (π / n))

Circumradius

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Circumradius = a / (2 × sin (π / n))

Selected Polygon Examples

Pentagon (5-Sided Polygon)

Interior Angle = 108°
Exterior Angle = 72°

Triangle

For area:

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Area = (base × height) / 2

Rectangle

Opposite sides equal
All interior angles = 90°

Parallelogram

Area formula:

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Area = base × height

Regular Polygon Calculator Tool

This tool is best for equiangular and equilateral polygons.

Benefits of Using the Calculator

Fast and accurate
Easy to use for both students and professionals
Avoid manual formula errors.

Use Cases for Students and Engineers

Geometry homework
CAD and architecture design
Math competitions and Olympiads
Construction layout planning

FAQs

What is a polygon properties calculator?

A tool that computes the sides, angles, perimeter, and area of polygons, both regular and irregular.

How accurate is this calculator?

Very accurate—it uses  standard and verified polygon formulas.

Can it calculate for irregular polygons?

Yes, though advanced outputs (like apothem) are easier with regular polygons.

What’s the difference between the incircle and the circumcircle?

Incircle: Touches all sides from inside.

Circumcircle: Passes through all vertices from outside.

Where can I use this calculator?

Visit hcalculators to access the full Polygon Properties Calculator Tool.

Final Thoughts

Whether you’re a student, engineer, or geometry enthusiast, the Polygon Properties Calculator on hcalculator simplifies your polygon calculations. From area and perimeter to radius and angles, it handles everything.

Shapes aren’t just sides and angles—they’re the foundation of how we understand space and design. Let hcalculator do the math, so you can focus on creativity and precision.