Polygon Calculator
Calculate area, perimeter, and angles of any regular polygon
Input Values
Results
Step-by-Step Solution
P = n × s
P = 6 × 10
P = 0.0000
A = (n × s²) / (4 × tan(π/n))
A = (6 × 10²) / (4 × tan(π/6))
A = 0.0000
Sum = (n - 2) × 180°
Sum = (6 - 2) × 180°
Sum = 0.00°
Each angle = Sum / n
Each angle = 0.00° / 6
Each angle = 0.00°
Understanding Polygons
What is a Polygon?
A polygon is a closed two-dimensional shape formed by straight line segments. A regular polygon has all sides of equal length and all angles equal. The word "polygon" comes from Greek: "poly" means many and "gon" means angle.
Key Formulas
Perimeter: P = n × s (where n = number of sides, s = side length)
Area: A = (n × s²) / (4 × tan(π/n))
Interior Angle Sum: Sum = (n - 2) × 180°
Each Interior Angle: Angle = (n - 2) × 180° / n
Central Angle: Central = 360° / n
Exterior Angle: Exterior = 360° / n
Common Regular Polygons
| Sides | Name | Each Interior Angle | Each Exterior Angle |
|---|---|---|---|
| 3 | Triangle | 60° | 120° |
| 4 | Square | 90° | 90° |
| 5 | Pentagon | 108° | 72° |
| 6 | Hexagon | 120° | 60° |
| 8 | Octagon | 135° | 45° |
| 10 | Decagon | 144° | 36° |
Frequently Asked Questions
What is a regular polygon?
A regular polygon is a polygon where all sides are equal in length and all interior angles are equal. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons.
How do you find the area of a regular polygon?
The area of a regular polygon can be calculated using the formula A = (n × s²) / (4 × tan(π/n)), where n is the number of sides and s is the side length. Alternatively, you can use A = (1/2) × perimeter × apothem.
What is the sum of interior angles of a polygon?
The sum of interior angles of any polygon is given by the formula (n - 2) × 180°, where n is the number of sides. For example, a triangle (3 sides) has an angle sum of 180°, while a square (4 sides) has 360°.
Why do exterior angles always sum to 360°?
The exterior angles of any polygon always sum to 360° because if you walk around the perimeter of a polygon, you make one complete turn (360°). This is true regardless of the number of sides or whether the polygon is regular or irregular.
What is the difference between interior and exterior angles?
An interior angle is formed inside the polygon at each vertex, while an exterior angle is formed by extending one side of the polygon. At each vertex, the interior and exterior angles are supplementary (they add up to 180°).
Can a polygon have 2 sides?
No, a polygon must have at least 3 sides. A two-sided figure would just be two line segments and wouldn't enclose an area. The minimum polygon is a triangle with 3 sides.
What is a central angle in a polygon?
The central angle is the angle formed at the center of a regular polygon by lines drawn to two consecutive vertices. For a regular polygon with n sides, the central angle is 360°/n.
How are polygons used in real life?
Polygons are everywhere in architecture (hexagonal tiles, octagonal stop signs), nature (honeycomb hexagons), engineering (pentagonal building designs), computer graphics (mesh polygons), and art. Understanding polygon properties is essential in many fields.
Related Calculators
Regular Polygon Calculator
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Pentagon Calculator
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Hexagon Calculator
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Octagon Calculator
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Interior Angle Calculator
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Polygon Area Calculator
Calculate area using multiple methods