Polygon Calculator
Calculate every property of a regular polygon online for free. Enter the number of sides to instantly get the interior angle, exterior angle, sum of interior angles, and number of diagonals. Add a side length to also compute area, perimeter, inradius, and circumradius — with a live polygon preview and full formula reference. Fast, private, and no signup required.
Enter the number of sides of a regular polygon to calculate interior angles, exterior angles, sum of interior angles, number of diagonals, and more. Optionally enter the side length to also compute area, perimeter, inradius, and circumradius. All calculations run locally in your browser — no signup required.
Whole number ≥ 3 (required)
Enter to calculate area, perimeter & radii
Why Use Our Polygon Calculator?
Instant Polygon Calculator Online
Enter the number of sides and get every polygon property in milliseconds — interior angles, exterior angles, sum of interior angles, diagonals, area, perimeter, inradius, and circumradius. The polygon calculator handles any regular polygon from a triangle (3 sides) up to 10,000 sides.
Secure Polygon Calculator — 100% Private
The polygon calculator runs entirely client-side in your browser. Your inputs are never uploaded to any server, stored, or tracked — completely private for academic, engineering, or design work.
Polygon Calculator Online — No Installation
Use the polygon calculator directly in any modern browser with no downloads, apps, or plugins required. Works on desktop, tablet, and mobile — 100% free forever with no signup required.
Full Polygon Analysis — Angles, Area, Radii & Diagonals
The polygon calculator goes beyond just angles. Enter an optional side length to unlock area, perimeter, inradius (apothem), and circumradius. A live SVG polygon preview updates instantly for shapes with 3–20 sides, and all formulas are shown for reference.
Common Use Cases for Polygon Calculator
School & University Geometry
Students use the polygon calculator to verify homework answers for interior angles, exterior angles, and diagonal counts. The polygon calculator shows all formulas alongside results, making it easy to understand the derivation and check work step by step.
Architecture & Construction
Architects and builders use the polygon calculator to design regular polygon floor plans, gazebos, towers, and decorative features. The area and perimeter outputs help estimate material quantities for any regular polygonal structure.
Engineering & CAD Design
Engineers use the polygon calculator to determine bolt-circle geometry, gear tooth profiles, and regular polygon cross-sections. The inradius and circumradius values are essential for fitting polygons inside or around circular constraints.
Graphic Design & Art
Designers use the polygon calculator to create precise regular polygon shapes for logos, patterns, and illustrations. The interior angle values ensure accurate vertex placement when drawing polygons programmatically or in vector tools.
Tiling & Tessellation
The polygon calculator helps determine which regular polygons can tile a plane — only triangles, squares, and hexagons tessellate perfectly. The interior angle output instantly shows whether a polygon's angles divide evenly into 360°.
Game Development & 3D Modelling
Game developers and 3D artists use the polygon calculator to generate regular polygon vertex coordinates for procedural geometry. The circumradius gives the radius of the circumscribed circle, which is the key parameter for placing vertices.
Understanding the Polygon Calculator
What is a Polygon Calculator?
A polygon calculator is a tool that computes the geometric properties of a regular polygon — a polygon where all sides are equal length and all interior angles are equal. Our online polygon calculator computes the sum of interior angles, each interior angle, each exterior angle, the number of diagonals, and — when a side length is provided — the area, perimeter, inradius (apothem), and circumradius. It supports any regular polygon from a triangle (3 sides) up to 10,000 sides, and runs entirely in your browser — no signup required.
How Our Polygon Calculator Works
- Enter the Number of Sides: Type any whole number ≥ 3 into the sides field, or click one of the quick-select preset buttons (Triangle, Square, Pentagon, Hexagon, Octagon, Decagon). The polygon calculator instantly computes all angle and diagonal properties.
- Optionally Enter a Side Length: To also calculate area, perimeter, inradius, and circumradius, enter the side length in the second field. The polygon calculator applies the standard regular polygon formulas using trigonometric functions for exact results.
- Review Results and Formulas: The polygon calculator displays all results in grouped stat cards, shows a live SVG polygon preview (for 3–20 sides), and lists every formula used — so you can verify the calculations or use them in your own work. Copy the full results summary with one click.
Key Polygon Formulas
- Sum of Interior Angles: (n − 2) × 180° — for a triangle (n=3) this gives 180°; for a hexagon (n=6) it gives 720°.
- Each Interior Angle: (n − 2) × 180° / n — for a regular hexagon this is 120°. As n → ∞, the interior angle approaches 180° (a circle).
- Each Exterior Angle: 360° / n — exterior angles always sum to 360° for any convex polygon. For a square: 360° / 4 = 90°.
- Number of Diagonals: n(n − 3) / 2 — a pentagon has 5 diagonals; a hexagon has 9. A triangle has 0 diagonals (no non-adjacent vertices).
- Area (regular polygon): (n × s²) / (4 × tan(π/n)) — where s is the side length. This formula uses the cotangent of the central half-angle.
- Inradius (apothem): s / (2 × tan(π/n)) — the perpendicular distance from the centre to the midpoint of any side. Used in area = ½ × perimeter × inradius.
- Circumradius: s / (2 × sin(π/n)) — the distance from the centre to any vertex. All vertices of a regular polygon lie on a circle of this radius.
Regular vs. Irregular Polygons
The polygon calculator computes properties of regular polygons — shapes where all sides are equal and all interior angles are equal. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons. An irregular polygon has sides or angles of different lengths, and its properties cannot be determined from the number of sides alone. The sum of interior angles formula (n − 2) × 180° applies to all convex polygons regardless of regularity, but the per-angle, area, and radius formulas in this polygon calculator assume a regular polygon.
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Frequently Asked Questions About Polygon Calculator
A polygon calculator computes the geometric properties of a regular polygon — interior angles, exterior angles, sum of interior angles, number of diagonals, area, perimeter, inradius, and circumradius. Our online polygon calculator handles any regular polygon from 3 to 10,000 sides and runs entirely in your browser with no signup required.
The interior angle of a regular n-sided polygon is (n − 2) × 180° / n. For a triangle (n=3): 60°. For a square (n=4): 90°. For a regular hexagon (n=6): 120°. The polygon calculator applies this formula instantly for any n.
A regular n-sided polygon has n(n − 3) / 2 diagonals. A triangle has 0, a quadrilateral has 2, a pentagon has 5, a hexagon has 9, and an octagon has 20. The polygon calculator computes this for any number of sides.
The inradius (apothem) is the perpendicular distance from the centre of the polygon to the midpoint of any side. The circumradius is the distance from the centre to any vertex. All vertices of a regular polygon lie on a circle of radius equal to the circumradius. The polygon calculator computes both when a side length is provided.
The polygon calculator is designed for regular polygons — where all sides and angles are equal. The sum of interior angles formula (n − 2) × 180° applies to all convex polygons, but the per-angle, area, and radius formulas assume regularity. For irregular polygons, you would need individual side lengths and angles.
The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides. For a regular polygon, each exterior angle is 360° / n. This is why a regular hexagon has exterior angles of 60° and a regular octagon has exterior angles of 45°.
Yes. The polygon calculator runs 100% locally in your browser. Your inputs are never sent to any server, stored, or tracked in any way — completely private for academic, engineering, or design work.
Yes — the polygon calculator is 100% free with no signup, no account, and no usage limits. Use it as many times as you need, completely free forever.
Only three regular polygons can tile a plane by themselves: equilateral triangles (interior angle 60°), squares (90°), and regular hexagons (120°). These are the only cases where the interior angle divides evenly into 360°. The polygon calculator's interior angle output makes it easy to check this for any polygon.