Regular Polygon Calculator
A side count and side length define a regular polygon in MiniWebtool’s Regular Polygon Calculator. Explore its area, perimeter, apothem, radii, angles, and diagonals with formulas and a matching diagram.
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Enter the number of sides and the side length below to get the area, perimeter, apothem, radii, angles and number of diagonals, with the steps.
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.
At a Glance
Regular Polygon Calculator finds the properties of a regular polygon from its number of sides and side length. It gives area, perimeter, apothem, circumradius, angles and diagonal count, with step-by-step formulas and a diagram.
| Feature | What you get | Good for |
|---|---|---|
| Inputs | Enter the number of sides, with a slider from 3 to 30, and the length of one side. | Any regular polygon from a triangle upward |
| Quick shapes | Buttons for triangle, square, pentagon, hexagon, octagon, decagon and dodecagon set the side count. | Common shapes such as a stop sign or hex nut |
| Measurements | Calculates area, perimeter, apothem (inradius), circumradius, interior angle, exterior angle, sum of angles and number of diagonals. | Geometry homework and design layouts |
| Formulas | Uses A = n x s squared / (4 tan(pi/n)), P = n x s, and the diagonal count n(n-3)/2, shown with the steps. | Checking each step of the working |
| Diagram overlays | An SVG diagram can show or hide the apothem, radius, diagonals and labels. | Seeing what apothem and circumradius mean |
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
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