Regular Polygon Calculator — Any Number of Sides

If you're dealing with a hexagonal shaft, a bolt head, or any part with a symmetric, multi-sided shape, you'll often need area, perimeter, angles, and apothem values on hand. This Regular Polygon Calculator gets you those numbers quickly from just the number of sides and the side length. Useful for mechanical layout, checking fits, or setting up CNC code. You'll find the formulas, a worked example, and some technical background further down the page.

What is a Regular Polygon?

A regular polygon has all sides of equal length and all interior angles equal. Examples include the equilateral triangle, square, hexagon, and octagon. This calculator gives you area, perimeter, interior angle, and apothem for any such shape, as long as it has at least 3 sides.

Simple Explanation

Think about cutting a part from stock where each edge is the same length and each angle is the same. That’s a regular polygon. As you add sides, the profile becomes more like a circle, and at high counts, the difference gets hard to see by eye. This tool handles the math so you can focus on whether the result is practical for your build.

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Regular Polygon Geometry Diagram

Regular Polygon Calculator

How to Use This Calculator

1. Enter the number of sides (n) — must be a whole number, 3 or greater.
2. Enter the side length (s) in your chosen unit.
3. Select Metric (mm) or Imperial (in) to match your project units.
4. Click Calculate to see your result.

📹 Video Walkthrough — How to Use This Calculator

Mathematical Equations

Simple Example

A regular hexagon (6 sides) with a side length of 10mm:

• Area: (6 × 10²) / (4 × tan(π/6)) = 259.81 mm²
• Perimeter: 6 × 10 = 60 mm
• Interior Angle: (6−2) × 180° / 6 = 120°
• Apothem: 10 / (2 × tan(π/6)) = 8.66 mm

Complete Technical Guide to Regular Polygons

Understanding Regular Polygon Geometry

A regular polygon is just a shape with identical sides and identical angles, and you’ll see them in everything from bolt heads and gear blanks to indexing plates. In a machine shop or on the design bench, knowing how to get precise measurements for these shapes speeds things up—especially when you want a tight fit or are managing tolerances in assemblies.

Every regular polygon breaks down nicely into n identical triangles, with the center as their common vertex. All the basic geometry—area, perimeter, angles—starts from this kind of breakdown.

Mathematical Foundation

The area formula A = (ns²)/(4 tan(π/n)) works because you’re just summing up n matching isosceles triangles that make up the shape. Each triangle’s base is the side length s; height is the apothem. The tangent function comes from basic trigonometry connecting the angles to these triangle dimensions.

Put another way: the apothem is s/(2 tan(π/n)), and each triangle’s area is (1/2) × base × height = (1/2) × s × apothem. Do that for all n sides and you have the complete area.

Engineering Applications

Hexagon heads on bolts or nuts are common because the 120° angle gives six easy gripping points with good strength. Honeycomb and hex-nut layouts use this shape to use less material without losing much strength. For routing or switching parts precisely, regular polygons show up as indexing plates—a 12-sided version gives twelve detents at 30° each, handy for controlled rotation and stops. Systems using actuators and rotary stages sometimes use these same polygons for setting up accurate angular steps.

Practical Design Considerations

When laying out regular polygon shapes, you’ll want to be clear about circumscribed versus inscribed dimensions. For instance, a hexagon inside a 10mm circle ends up at 10mm across the flats, but if it surrounds a 10mm circle, the sides bump up to about 11.55mm. These are easy mistakes to make when tolerance is tight.

Corners concentrate stress—especially if sides are short (few-sided polygons). Adding a small fillet at each vertex will spread loads and reduce cracking, especially for drive shafts or connectors. As you add more sides, the shape behaves more like a round section and stress at corners drops.

Worked Example: Hexagonal Shaft Design

Here's a quick hex shaft example used for torque transmission:

Given: Regular hexagon, side length s = 15mm

Calculations:

• Number of sides: n = 6
• Area: A = (6 × 15²)/(4 tan(π/6)) = (6 × 225)/(4 × 0.5774) = 1350/2.309 = 584.8 mm²
• Perimeter: P = 6 × 15 = 90mm
• Interior angle: θ = (6-2) × 180°/6 = 120°
• Apothem: a = 15/(2 tan(π/6)) = 15/1.155 = 12.99mm

This tells you the shaft will have 584.8 mm² of usable cross-sectional area. The apothem here (12.99mm) is basically the largest bore or shaft you could inscribe inside the hex and still have it touch all sides—handy for checking fits and machining clearances.

Advanced Applications in Automation

Regular polygons are useful for cam profiles, indexing tables, and gear blanks in automation. If you need an automated fixture with eight stopped positions, use an octagon—each face gives you a home position at 45°. The math also helps for robotics if you need to plan precise, repeatable paths; the symmetry makes both programming and construction more manageable.

Manufacturing and Tolerance Considerations

Even a 0.1mm tolerance on a cut polygon side can vary the area quite a bit, especially for shapes with fewer sides. If you’re CNC programming, use the calculator’s output to double-check minimum tool diameter versus the apothem, and consider how side length changes with any tool wear or measurement error.

Quality checks on CMMs often use the polygon’s calculated angles for spot checks and part inspection. Reference the calculated angles and distances to make sure the cut or milled part is actually regular—not just visually close.

Regular polygon formulas are basic, but they come up again and again if you work hands-on in fabrication, assembly, or design. Whether you’re trying to maximize space, plan rotation stops, or just ensure reliable wrench fits, being able to crunch these numbers quickly saves time and helps prevent mistakes when tolerances matter.

Frequently Asked Questions

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