Enter the number of sides and side length — get the area, perimeter, interior angle, and apothem instantly.
Enter the number of sides and side length to see the results
This regular polygon calculator finds the area, perimeter, interior angle, and apothem of any regular polygon — a shape with equal sides and equal angles — from just the number of sides (n) and the side length (s). From a triangle (n=3) to a dodecagon (n=12) and beyond, the same set of formulas applies to every regular polygon, using basic trigonometry on the angle each side subtends at the center. Whether you're using this as a regular polygon area calculator for a hexagonal paver, checking the interior angle of a polygon formula for homework, or sizing a stop-sign-shaped sign, this free tool computes every related value instantly.
Given the number of sides (n) and side length (s), the calculator computes perimeter (n × s), area using the tangent-based formula (n × s²) / (4 × tan(π/n)), the interior angle at each vertex ((n−2) × 180 / n), and the apothem — the distance from the center to the midpoint of a side — using s / (2 × tan(π/n)).
This tool suits geometry students studying polygon properties, designers and fabricators working with hexagonal, octagonal, or other regular-polygon components, sign-makers sizing stop signs and other regulated shapes, and anyone with a side count and side length who needs the area, perimeter, angle, or apothem without manually applying trigonometry.
Regular polygons show up constantly in engineering and design because their symmetry makes them efficient to manufacture, tile, and structurally analyze — hex bolts, honeycomb structures, and floor tiles all rely on precise regular-polygon geometry. As the number of sides grows, a regular polygon's area formula smoothly approaches a circle's area formula, which is a classic illustration of how π itself can be understood as a limiting case.
Manufacturers compute hex bolt or nut head dimensions from side length and count. Sign-makers size regulated shapes like the regular-octagon stop sign. Architects and builders plan gazebo, pavilion, or floor-tile layouts using regular polygons. Packaging designers compute regular polygon base areas for containers. Students and teachers use it to verify polygon geometry and trigonometry homework quickly.
How the number of sides and side length determine every other property
The same four formulas work for any regular polygon — triangle, square, pentagon, hexagon, or beyond — just by changing n.
A regular polygon splits into n identical isosceles triangles from its center, which is the geometric basis for both the area and apothem formulas.
As the number of sides increases without bound, a regular polygon's area and perimeter formulas converge toward a circle's πr² and 2πr.
From entering the sides and length to reading all four results
Determine how many equal sides the polygon has (n), a whole number of 3 or more.
Find the length of one side (s), since all sides are equal in a regular polygon.
Type n and s into the two input fields.
The calculator computes perimeter, area, interior angle, and apothem using standard regular-polygon formulas.
Review the perimeter, area, interior angle, and apothem shown together in the results panel.
Solving a regular hexagon with side length 4
A hexagonal paver has 6 equal sides, each measuring 4 cm. What are its perimeter, area, interior angle, and apothem?
Explanation: Note that a regular hexagon's area can also be cross-checked as (Perimeter × Apothem) ÷ 2 = (24 × 3.4641) ÷ 2 ≈ 41.5692 — matching the tangent-based formula exactly, since both describe the same n congruent triangles from the center.
What each output represents and how they relate
| Property | What It Represents | Unit Type |
|---|---|---|
| Perimeter | Total distance around all n sides | Linear (same unit as side length) |
| Area | Two-dimensional space enclosed by the polygon | Squared (e.g. cm², m², in²) |
| Interior Angle | The angle at each vertex, inside the polygon | Degrees |
| Apothem | Distance from center to the midpoint of a side | Linear (same unit as side length) |
Reading the results together: perimeter and area describe the polygon's overall size, the interior angle describes its "sharpness" at each corner, and the apothem — closely tied to area — describes how far the center sits from each edge.
Typical ranges: interior angles range from 60° (triangle, n=3) up toward but never reaching 180° as n grows large; as n grows very large, a regular polygon visually and numerically approximates a circle, with the apothem approaching the circle's radius.
Manual verification: multiply n and s for perimeter directly; recompute (n−2) × 180 ÷ n for the interior angle; and check that Perimeter × Apothem ÷ 2 reproduces the same area as the tangent-based formula.
Where computing regular polygon area, perimeter, angles, and apothem is genuinely useful
Check regular polygon area, perimeter, and interior angle problems step by step.
Compute cell area and packing dimensions for honeycomb-style hexagonal layouts.
Verify hex bolt head or nut dimensions from side length and side count.
Size the regular-octagon stop sign shape to required dimensions.
Plan hexagonal or octagonal gazebo floor area and framing length.
Calculate hexagonal or pentagonal paver dimensions for patios and floors.
Verify regular-polygon coin or medallion area and edge dimensions.
Check regular-polygon support frame geometry for circular attractions.
Plan many-sided polygon approximations used in dome and rotunda construction.
Compute precise regular-polygon proportions for design motifs and icons.
Size regular-polygon container or box bases for consistent volume.
Verify regular-polygon component dimensions in technical drawings.
Determine regular-polygon facet proportions for gem or pendant design.
What this regular polygon calculator does well, and where it doesn't apply
How interior angle and shape change as the number of sides increases, using side length = 4 as a reference
| Polygon (n) | Interior Angle | Perimeter (s=4) | Area (s=4) |
|---|---|---|---|
| Triangle (3) | 60° | 12 | ≈6.9282 |
| Square (4) | 90° | 16 | 16 |
| Pentagon (5) | 108° | 20 | ≈27.5276 |
| Hexagon (6) | 120° | 24 | ≈41.5692 |
| Octagon (8) | 135° | 32 | ≈77.2549 |
Summary: This regular polygon calculator gives you an instant, free way to find perimeter, area, interior angle, and apothem for any regular polygon, with every formula shown. Pair it with related tools like the Circle Calculator and Area Calculator for a fuller picture of polygon and curved-shape geometry.
Common questions about regular polygon measurements
Trusted educational references to go deeper on polygons and trigonometry
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