Regular Polygon Calculator
Calculate the area, perimeter, apothem, circumradius, interior angles, exterior angles, and number of diagonals of any regular polygon. Enter the number of sides and side length to get instant results with step-by-step formulas and an interactive diagram.
Your ad blocker is preventing us from showing ads
MiniWebtool is free because of ads. If this tool helped you, please support us by going Premium (ad‑free + faster tools), or allowlist MiniWebtool.com and reload.
- Allow ads for MiniWebtool.com, then reload
- Or upgrade to Premium (ad‑free)
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
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
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
You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.
Related MiniWebtools:
Geometry Calculators:
- Arc Length Calculator Featured
- Cartesian to Polar Coordinates Converter
- Circle Calculator
- Distance Between Two Points Calculator
- Ellipse Circumference Calculator
- General Triangle Solver
- Golden Rectangle Calculator
- Golden Section Calculator
- Hypotenuse Calculator Featured
- Midpoint Calculator
- Polar to Cartesian Converter
- Pythagorean Theorem Calculator
- Rectangle Calculator
- Slope Calculator
- Slope Intercept Form Calculator
- Square Calculator
- Shoelace Formula Calculator New
- Triangle Centroid Calculator New
- Triangle Orthocenter Calculator New
- Point to Plane Distance Calculator New
- Sphere Equation Calculator New
- Cone Flat Pattern (Template) Generator New
- Polygon Diagonals Calculator New
- Euler Characteristic Calculator New
- Point-Slope Form Calculator New
- Equation of a Line Calculator New
- Parallel and Perpendicular Line Calculator New
- Parabola Calculator New
- Hyperbola Calculator New
- Conic Section Identifier New
- Regular Polygon Calculator New
- Irregular Polygon Area Calculator New
- Frustum Calculator New
- Torus Calculator New
- 3D Distance Calculator New
- Great Circle Distance Calculator New
- Circumscribed Circle (Circumcircle) Calculator New
- Inscribed Circle (Incircle) Calculator New
- Angle Bisector Calculator New
- Tangent Line to Circle Calculator New
- Heron's Formula Calculator New
- Coordinate Geometry Distance Calculator New
- Parametric Curve Grapher New
- Parametric Curve Grapher New