Apothem Calculator
Calculate the apothem of regular polygons from side length or radius
Input Values
Results
Step-by-Step Solution
a = s / (2 × tan(π/n))
a = 10 / (2 × tan(π/6))
a = 0.0000
A = (1/2) × perimeter × apothem
A = (1/2) × 0.0000 × 0.0000
A = 0.0000
Understanding the Apothem
What is an Apothem?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It's also called the inradius because it's the radius of the largest circle that fits inside the polygon (the incircle).
Apothem Formulas
From Side Length: a = s / (2 × tan(π/n))
From Radius: a = r × cos(π/n)
Relationship: a = r × cos(π/n)
Area Formula: A = (1/2) × perimeter × apothem
Why is the Apothem Important?
- Essential for calculating polygon area using A = (1/2) × P × a
- Represents the inradius (radius of inscribed circle)
- Forms the height of triangular sections when dividing polygon from center
- Used in engineering for calculating clearances and fits
Frequently Asked Questions
What is the difference between apothem and radius?
The radius (circumradius) extends from the center to a vertex, while the apothem (inradius) is the perpendicular distance from the center to the midpoint of a side. The apothem is always shorter than the radius.
How do you find the apothem?
From side length: a = s / (2 × tan(π/n)). From radius: a = r × cos(π/n). You can also calculate it trigonometrically as the altitude of the isosceles triangles formed by connecting the center to adjacent vertices.
Why is it called apothem?
The word "apothem" comes from Greek "apo" (from) + "tithenai" (to place), literally meaning "to place off from." It refers to the perpendicular distance from the center.
Can irregular polygons have an apothem?
The apothem is specifically defined for regular polygons where all sides are equal. Irregular polygons don't have a single apothem value since the perpendicular distances from center to different sides vary.
How is the apothem used in area calculations?
The area formula A = (1/2) × perimeter × apothem works because a regular polygon can be divided into n congruent triangles, each with base = side length and height = apothem.
What polygon has apothem equal to radius?
No regular polygon has apothem equal to radius. As the number of sides increases, the ratio a/r approaches 1, approaching a circle. For a circle, the "apothem" and radius are equal.