Polygon Area Calculator
Find the area and perimeter of any regular polygon. Enter the number of sides and the side length, and this tool handles the trigonometry.
Understanding Regular Polygon Area
A regular polygon is perfectly symmetric. All sides have the same length, and all interior angles are equal. This symmetry allows a single formula to compute the area based only on the number of sides and the side length.
The formula A = (n × s²) / (4 × tan(π/n)) comes from dividing the polygon into n congruent isosceles triangles, each with its apex at the center. The area of one triangle is (s × a) / 2, where a is the apothem. Multiply by n triangles, and you get (n × s × a) / 2, which simplifies to (P × a) / 2.
The trigonometric version eliminates the need to measure the apothem. The tangent function relates the side length and the central angle to the apothem, allowing the formula to work from just two inputs: n and s.
Real-World Uses of Regular Polygons
Regular polygons appear in engineering, architecture, and design. Hexagonal nuts and bolts use six-sided symmetry for efficient wrenching. Honeycomb structures exploit hexagons because they tile perfectly with no gaps, maximizing strength while minimizing material.
In architecture, octagonal and hexagonal floor plans create unique spaces. The symmetry distributes loads evenly, making the structure stable. Knowing the area helps with material estimation and space planning.
Game boards, tile patterns, and decorative designs often feature regular polygons. Pentagons and octagons create visual interest that squares and triangles cannot. Calculating the area ensures consistent proportions and proper fit.
Special Cases: Triangles, Squares, and Hexagons
An equilateral triangle is a regular 3-sided polygon. The area formula simplifies to A = (s² √3) / 4. This is a well-known result that emerges from the general polygon formula when n = 3.
A square is a regular 4-sided polygon. The formula reduces to A = s², the simplest case. The tangent term tan(π/4) equals 1, so the denominator becomes 4, and the numerator becomes 4s², giving s².
A regular hexagon (6 sides) has area A = (3√3 / 2) × s². This formula is popular in tiling and material science because hexagons pack efficiently. The calculator handles these and all other regular polygons with the same general formula.
Frequently Asked Questions
What is a regular polygon?
A regular polygon has all sides equal in length and all interior angles equal. Examples include equilateral triangles, squares, regular pentagons, and hexagons.
What is the formula for the area of a regular polygon?
Area equals (n × s²) / (4 × tan(π/n)), where n is the number of sides and s is the side length. Alternatively, A = (perimeter × apothem) / 2 if you know the apothem.
What is an apothem?
The apothem is the distance from the center of the polygon to the midpoint of any side. It is the radius of the inscribed circle and is perpendicular to the side.
How do I find the perimeter of a regular polygon?
Multiply the side length by the number of sides: P = n × s. Since all sides are equal, perimeter is straightforward.
Does this formula work for irregular polygons?
No. This formula assumes all sides and angles are equal. Irregular polygons require different methods, such as dividing into triangles or using coordinate geometry.