Apothem Calculator

To find the apothem of a regular polygon, divide the side length ( $S$ ) by $2 \cdot \tan(180/n)$ , where $n$ is the number of sides.

The Apothem Calculator is a handy tool for determining the apothem (the perpendicular distance from the center to a side) of a regular polygon.

The apothem is essential for calculating the area of polygons, such as hexagons, pentagons, and octagons, and plays a vital role in geometry and design applications.

Formula:

$a = S / (2 \cdot \tan(180/n))$

Variable	Description	Unit
$a$	Apothem length	Any length unit
$S$	Side length of the polygon	Same as $a$
$n$	Number of sides	Dimensionless

Solved Calculations:

Example 1: Calculate the Apothem of a Hexagon ( $S = 10$ , $n = 6$ )

Step	Value	Explanation
Side Length ( $S$ )	$10$	Given
Number of Sides ( $n$ )	$6$	Given
Calculation	$a = 10 / (2 \cdot \tan(180/6))$	Substitute values into the formula
Simplification	$a = 10 / (2 \cdot \tan(30))$	Divide by $2 \cdot \tan(30)$
Result	$a = 8.66$	Apothem length

Example 2: Calculate the Apothem of a Pentagon ( $S = 8$ , $n = 5$ )

Step	Value	Explanation
Side Length ( $S$ )	$8$	Given
Number of Sides ( $n$ )	$5$	Given
Calculation	$a = 8 / (2 \cdot \tan(180/5))$	Substitute values into the formula
Simplification	$a = 8 / (2 \cdot \tan(36))$	Divide by $2 \cdot \tan(36)$
Result	$a = 5.52$	Apothem length

What is the Apothem Calculator?

The Apothem Calculator is a practical tool that helps calculate the apothem of regular polygons, a critical element in geometry used to determine the area of polygons.

The apothem is the perpendicular distance from the center of a polygon to the midpoint of one of its sides, making it essential for solving various geometric problems.

This calculator supports polygons of different shapes, including hexagons, pentagons, squares, and decagons. By inputting known values such as the side length, number of sides, or radius, users can quickly find the apothem and calculate the area of regular polygons.

For example, if the side length of a hexagon is provided, the calculator determines the apothem using the appropriate formula and even extends the calculation to the area.

The tool also offers step-by-step explanations, making it particularly useful for students and educators. It simplifies solving equations like the apothem of an equilateral triangle or determining the apothem of a decagon, ensuring accurate results for both academic and practical applications.

Final Words

The Apothem Calculator is an essential resource for calculating the apothem and area of regular polygons effortlessly. It enhances precision, saves time, and supports a wide range of geometric calculations, making it invaluable for students, teachers, and professionals alike.