Regular Polygon Calculator

Calculate the area, perimeter, apothem, circumradius, interior and exterior angles and number of diagonals of any regular polygon. Enter the number of sides and side length for results with step-by-step formulas and a diagram.

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About Regular Polygon Calculator

The Regular Polygon Calculator computes all geometric properties of a regular polygon given the number of sides and the side length. A regular polygon has all sides equal in length and all interior angles equal in measure. This calculator instantly determines the area, perimeter, apothem (inradius), circumradius, interior angle, exterior angle, sum of angles, and number of diagonals, with step-by-step formulas and an interactive SVG diagram.

Common Regular Polygons

△

Triangle

3 sides · 60°

◇

Square

4 sides · 90°

⬠

Pentagon

5 sides · 108°

⬡

Hexagon

6 sides · 120°

⯃

Octagon

8 sides · 135°

◯

Decagon

10 sides · 144°

Key Formulas for Regular Polygons

For a regular polygon with n sides and side length s, these formulas apply:

Property	Formula	Description
Perimeter	\(P = n \times s\)	Total length of all sides
Interior Angle	\(\frac{(n-2) \times 180°}{n}\)	Angle at each vertex
Exterior Angle	\(\frac{360°}{n}\)	Supplement of interior angle
Apothem	\(a = \frac{s}{2\tan(\pi/n)}\)	Center to midpoint of side
Circumradius	\(R = \frac{s}{2\sin(\pi/n)}\)	Center to vertex
Area	\(A = \frac{n \times s^2}{4\tan(\pi/n)}\)	Enclosed surface area
Diagonals	\(d = \frac{n(n-3)}{2}\)	Number of diagonal lines

Understanding Apothem vs. Circumradius

The apothem (also called the inradius) is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle. The circumradius is the distance from the center to any vertex and is the radius of the circumscribed circle. The relationship between them is: \(R^2 = a^2 + (s/2)^2\), where s is the side length. As the number of sides increases, the apothem approaches the circumradius, and both approach the radius of a circle.

How to Use the Regular Polygon Calculator

Choose the number of sides: Enter a number (3 or more) in the "Number of Sides" field, or use the slider for quick selection. You can also click a quick example button like Pentagon, Hexagon, or Octagon.
Enter the side length: Type the length of one side of the polygon.
Click Calculate: Press the "Calculate Polygon" button to compute all properties.
Review the results: See the area, perimeter, apothem, circumradius, interior angle, exterior angle, diagonals count, step-by-step formulas, and interactive SVG diagram.
Explore the diagram: Toggle the Apothem, Radius, Diagonals, and Labels overlays to visualize different geometric features.

Practical Applications of Regular Polygons

Regular polygons appear everywhere in architecture, engineering, and nature. Stop signs are regular octagons. Hex nuts and bolts use hexagonal shapes for optimal grip. Soccer balls combine regular pentagons and hexagons. Honeycomb cells are regular hexagons because they tile the plane with minimal material. In architecture, polygonal floor plans and domed structures use regular polygon geometry for structural stability and aesthetic appeal.

Regular Polygons and Circles

As the number of sides of a regular polygon increases, the shape approaches a circle. Both the apothem and circumradius converge to the same value (the circle's radius), and the area approaches \(\pi r^2\). Ancient mathematicians like Archimedes used inscribed and circumscribed regular polygons to approximate the value of \(\pi\). A regular 100-gon already closely resembles a circle to the naked eye.

FAQ

What is a regular polygon?

A regular polygon is a polygon with all sides of equal length and all interior angles of equal measure. Examples include equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), and regular hexagons (6 sides). The more sides a regular polygon has, the closer it approximates a circle.

How do you calculate the area of a regular polygon?

The area of a regular polygon with n sides of length s is calculated using the formula A = (n × s²) / (4 × tan(π/n)). An equivalent formula is A = (1/2) × perimeter × apothem. Both give the same result.

What is the apothem of a regular polygon?

The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It is also the radius of the inscribed circle (the largest circle that fits inside the polygon). The formula is a = s / (2 × tan(π/n)).

How many diagonals does a regular polygon have?

A regular polygon with n sides has n(n−3)/2 diagonals. For example: a triangle has 0 diagonals, a square has 2, a pentagon has 5, a hexagon has 9, and an octagon has 20.

What is the difference between apothem and circumradius?

The apothem (inradius) is the distance from the center to the midpoint of a side, while the circumradius is the distance from the center to any vertex. The circumradius is always larger than the apothem. They are related by R² = a² + (s/2)², where s is the side length.

Reference this content, page, or tool as:

"Regular Polygon Calculator" at https://MiniWebtool.com/regular-polygon-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-02

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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