Polygon Calculator

Free Online Polygon Calculator

This polygon calculator helps you instantly find the area, perimeter, apothem, circumradius, and interior and exterior angles of any regular polygon. Just enter the number of sides and the side length, and the calculator computes every measurement automatically, complete with a fully labeled diagram and a step-by-step solution showing exactly how each formula was applied.

Whether you're a student working through geometry and mensuration homework, a teacher preparing worked examples, a designer laying out tiles or signage, or simply someone who needs a fast and accurate area of a polygon calculator or interior angle calculator, this tool covers every common regular-polygon calculation you're likely to need — from a triangle and square all the way up to a dodecagon.

A regular polygon is a flat, closed shape with all sides equal in length and all interior angles equal in measure — think of an equilateral triangle, a square, a regular pentagon, or a regular hexagon. Because every side and angle is identical, only two values are needed to define the entire shape: the number of sides (n) and the length of one side (s). This calculator uses exactly those two inputs to solve everything else.

Polygon Formulas Used

A regular polygon with n sides, each of length s, follows these standard mensuration formulas:

• Perimeter: P = n × s (the side length multiplied by the number of sides)
• Apothem: ap = s / (2 × tan(π/n)) (the perpendicular distance from the center to the midpoint of a side)
• Area: A = (n × s²) / (4 × tan(π/n)), which is the same as A = (1/2) × P × ap
• Circumradius: R = s / (2 × sin(π/n)) (the distance from the center to any vertex)
• Interior angle: ((n − 2) × 180) / n degrees
• Exterior angle: 360 / n degrees
• Sum of interior angles: (n − 2) × 180 degrees

How to Use This Polygon Calculator

Using this calculator only takes two steps. First, enter the number of sides of your regular polygon — for example, 3 for a triangle, 6 for a hexagon, or 8 for an octagon. Second, enter the length of one side. The calculator instantly displays the area, perimeter, apothem, circumradius, and both interior and exterior angles, along with a labeled diagram that shows the side length, apothem, and circumradius drawn directly on the shape.

This means you never have to manually apply the regular polygon area formula or work out interior angles yourself — the calculator handles all the trigonometry instantly, no matter how many sides your shape has. It's especially useful for tiling and paving layouts, sign and logo design, and any geometry problem involving regular shapes beyond the basic triangle, square, and circle.

Worked Example

Suppose you're designing a regular hexagonal paving tile with a side length of 5 centimeters. Enter n = 6 and s = 5. The calculator first finds the apothem (ap = 5 / (2 × tan(π/6)) ≈ 4.33 centimeters), then the area (A = (6 × 5²) / (4 × tan(π/6)) ≈ 64.95 square centimeters), and the perimeter (P = 6 × 5 = 30 centimeters). It also reports the interior angle (((6 − 2) × 180) / 6 = 120°) and the exterior angle (360 / 6 = 60°).

That tells you each tile covers about 64.95 square centimeters, needs 30 centimeters of edge trim, and meets its neighbors at a clean 120-degree interior angle — exactly why hexagonal tiles fit together perfectly with no gaps.

Apothem vs. Circumradius: What's the Difference?

It's easy to mix up these two measurements, but the diagram above makes the difference clear. The apothem (shown as the dashed green line) runs from the center of the polygon straight to the midpoint of one of its sides — it's always shorter, since it's a perpendicular distance to a flat edge. The circumradius (shown as the dashed purple line) runs from the center to one of the vertices (corners) — it's always longer, since it reaches all the way to a pointed corner rather than stopping at an edge. Both distances are needed for different formulas: the apothem drives the area calculation, while the circumradius describes the size of the circle that would pass through every corner of the polygon.

Common Regular Polygons by Number of Sides

Regular polygons are usually named after their number of sides. Here are the most common ones you'll come across in geometry and everyday design, all of which this calculator supports:

• 3 sides — Equilateral triangle (60° interior angles)
• 4 sides — Square (90° interior angles)
• 5 sides — Regular pentagon (108° interior angles)
• 6 sides — Regular hexagon (120° interior angles)
• 7 sides — Regular heptagon (≈128.57° interior angles)
• 8 sides — Regular octagon (135° interior angles)
• 9 sides — Regular nonagon (140° interior angles)
• 10 sides — Regular decagon (144° interior angles)
• 12 sides — Regular dodecagon (150° interior angles)

Regular vs. Irregular Polygons

It's worth noting the distinction this calculator relies on: a regular polygon has all sides equal and all interior angles equal, which is what allows the entire shape to be described with just two numbers, n and s. An irregular polygon, by contrast, can have sides and angles of completely different sizes — think of an arrow shape, an L-shaped room, or a random hand-drawn quadrilateral. As the number of sides in a regular polygon increases, the shape gets closer and closer to a perfect circle, with the apothem approaching the circumradius and the interior angle approaching 180°. This is the same underlying idea used in calculus to approximate the area of a circle using polygons with more and more sides.

Where Polygon Calculations Are Used

Regular polygon geometry and mensuration formulas appear constantly in real life and across many fields of study:

• Students and teachers use it for geometry and mensuration homework, competitive exams, and lesson planning.
• Architects and engineers use polygon measurements when designing hexagonal or octagonal rooms, gazebos, and structural panels.
• Tiling, paving, and flooring professionals rely on regular polygon area and angle calculations to plan seamless tile layouts.
• Graphic and product designers use polygon geometry for logos, icons, packaging shapes, and honeycomb-style patterns.
• Manufacturing and machining industries calculate polygon dimensions for bolt heads, nuts, gears, and other hexagonal or octagonal components.