Number of Sides
Known Measurement
Your Results
Results are also pinned to the bottom of your screen.Regular Polygon Properties
Area
93.5307
square units
Perimeter
36
units
Formulas Used
Area = (n × s²) / (4 × tan(π/n)) = (6 × 6²) / (4 × tan(0.5236)) = 93.5307Perimeter = n × s = 36Interior angle = (n − 2) × 180° / n = (6 − 2) × 180° / 6 = 120°Apothem = s / (2 × tan(π/n)) = 5.1962
Quick Answer
To calculate regular polygon properties with n sides of length s: Area = (n x s^2) / (4 x tan(pi/n)). Perimeter = n x s. Interior angle = (n - 2) x 180 / n degrees. Apothem = s / (2 x tan(pi/n)). For a regular hexagon with side 6: Area = 93.53 square units, Perimeter = 36 units, Interior angle = 120 degrees, Apothem = 5.20 units.
Key Facts
- Formula: Area = (n x s^2) / (4 x tan(pi/n))
- Formula: Interior angle = (n - 2) x 180 / n degrees
- Formula: Apothem = s / (2 x tan(pi/n))
- A regular hexagon with side 6 has area 93.53 and perimeter 36
- As n grows, a regular polygon approaches a circle of the same circumradius
- The apothem is the distance from center to the midpoint of any side
- Area from circumradius R: Area = (n/2) x R^2 x sin(2 x pi/n)
- The sum of interior angles of any polygon is (n - 2) x 180 degrees
Frequently Asked Questions
How do I calculate the area of a regular polygon?
Use Area = (n x s^2) / (4 x tan(pi/n)), where n is the number of sides and s is the side length. For a hexagon (n = 6) with side 6: Area = (6 x 36) / (4 x tan(30 degrees)) = 216 / 2.309 = 93.53 square units. Equivalently, Area = (1/2) x perimeter x apothem.
What is the apothem of a polygon?
The apothem is the perpendicular distance from the center to the midpoint of any side. It equals s / (2 x tan(pi/n)). For a hexagon with side 6, the apothem is 6 / (2 x 0.5774) = 5.20 units. Area = (1/2) x apothem x perimeter.
What is the interior angle of a regular polygon?
Interior angle = (n - 2) x 180 / n degrees. Triangles: 60 degrees. Squares: 90. Pentagons: 108. Hexagons: 120. The sum of all interior angles is always (n - 2) x 180 degrees, and each exterior angle is 360 / n.
How do I calculate a polygon from the circumradius?
The circumradius R is the distance from center to any vertex. Side length s = 2 x R x sin(pi/n), then use the standard formulas. For area directly: Area = (n/2) x R^2 x sin(2 x pi/n).
Why does a hexagon tile perfectly but a pentagon does not?
Regular polygons tile a plane only when their interior angle divides 360 evenly. The hexagon angle is 120 degrees (360 / 120 = 3), so three fit around a point. The pentagon angle is 108 degrees, which leaves a 36-degree gap.
What is a regular polygon?
A polygon where all sides are equal and all interior angles are equal. Examples: equilateral triangle (3), square (4), regular pentagon (5), regular hexagon (6), up to the regular dodecagon (12). This calculator covers n = 3 through 12.