Area of a Regular Polygon Calculator

A regular polygon is a two-dimensional shape with equal-length sides and equal interior angles. Depending on the variables you have available, the calculator uses one of three fundamental geometric formulas. When you know the number of sides (n) and the side length (a), the area is derived using the equation: Area = (n * a^2) / (4 * tan(pi / n)). If you are given the apothem (apo), which is the line segment from the center perpendicular to a side, the formula is Area = apo^2 * n * tan(pi / n). Finally, if you know the circumradius (r), the distance from the center to a vertex, the formula used is Area = 0.5 * r^2 * n * sin(2 * pi / n).

Let us calculate the area of a regular pentagon (a polygon with 5 sides) where each side length (a) measures 3 units. First, identify the number of sides, n = 5, and the side length, a = 3. Substitute these values into the side-length formula: Area = (5 * 3^2) / (4 * tan(pi / 5)). Next, square the side length to get 9, multiplying it by 5 yields 45. For the denominator, calculate tan(pi / 5), which is approximately 0.7265, and multiply by 4 to get 2.9061. Finally, divide 45 by 2.9061 to find an area of approximately 15.48 square units.

Always double-check your angle measurement settings, as trigonometric functions in these formulas require radians or degrees depending on how your input processor reads them. Ensure that all linear dimensions use the exact same unit of measurement before calculating, converting inches to feet or centimeters to meters as necessary. When working with apothems or circumradii, remember that precise internal angle divisions are critical for accurate multi-decimal approximations.