Polygon Calculator
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A regular polygon with n sides of length s has area A = ns²/(4 tan(180°/n)) and each interior angle measures (n − 2) × 180° ÷ n. A regular hexagon with side 4 has area ≈ 41.57 and 120° angles.
The dashed circle is the circumscribed circle (radius R).
| Measurements | |
| Side length s | 4 |
| Apothem a (center to side) | 3.464101615 |
| Circumradius R (center to vertex) | 4 |
| Perimeter P | 24 |
| Area A | 41.56921938 |
| Angles | |
| Each interior angle | 120° |
| Sum of interior angles | 720° |
| Each exterior angle | 60° |
| Central angle | 60° |
| Counts | |
| Number of diagonals n(n − 3)/2 | 9 |
| Name | regular hexagon |
Show steps
- You gave the side length s = 4 for a polygon with n = 6 sides.
- Apothem: \(a = \frac{s}{2\tan(180^\circ/n)}\) = 3.4641016. Circumradius: \(R = \frac{s}{2\sin(180^\circ/n)}\) = 4.
- Area: \(A = \frac{1}{2}Pa\) = ½ × 24 × 3.4641016 = 41.569219.
- Interior angle: (n − 2) × 180° ÷ n = 120°. Exterior angle: 360° ÷ n = 60°.
How to use the polygon calculator
- Choose Regular polygon, enter the number of sides, and give any one measurement: side, apothem, circumradius, area or perimeter.
- Or choose Any polygon from vertex coordinates and list the corners in order.
- Press Calculate to get every measurement, the angles and a drawing.
Regular polygon formulas
| Measure | Formula |
|---|---|
| Interior angle | (n − 2) × 180° ÷ n |
| Exterior angle | 360° ÷ n |
| Apothem | a = s ÷ (2 tan(180°/n)) |
| Circumradius | R = s ÷ (2 sin(180°/n)) |
| Area | A = ½ × perimeter × apothem |
| Diagonals | n(n − 3) ÷ 2 |
Area of any polygon: the shoelace formula
List the vertices (x₁, y₁), …, (xₙ, yₙ) in order around the shape. Then \(A = \frac{1}{2}\left|\sum_{i=1}^{n} (x_{i}y_{i+1} – x_{i+1}y_{i})\right|\), wrapping back to the first point. For the rectangle (0, 0), (4, 0), (4, 3), (0, 3): ½ |0 + 12 + 12 + 0| = 12.
Polygon names
| Sides | Name | Interior angle (regular) |
|---|---|---|
| 3 | triangle | 60° |
| 4 | quadrilateral (square) | 90° |
| 5 | pentagon | 108° |
| 6 | hexagon | 120° |
| 8 | octagon | 135° |
| 10 | decagon | 144° |
| 12 | dodecagon | 150° |
Polygon practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you find the area of a regular polygon?
Multiply half the perimeter by the apothem: A = ½Pa. Without the apothem, use A = ns²/(4 tan(180°/n)).
How do you find the area of an irregular polygon?
Use the shoelace formula with the coordinates of its vertices, or split it into triangles and rectangles.
What is the sum of the interior angles of a polygon?
(n − 2) × 180°, where n is the number of sides.
Does the order of the vertices matter?
Yes. List them in order around the boundary. Jumping across the shape makes the sides cross and gives a wrong area; the calculator warns you.
What is an apothem?
The distance from the center of a regular polygon to the middle of a side. It is the radius of the inscribed circle.
Sources
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