Apothem

Mathematical concept in geometry

Apothem

The apothem of a regular polygon is the line segment from the center of the polygon to the midpoint of one of its sides. It is perpendicular to the side and is a key element in various geometric calculations, particularly in determining the area of a regular polygon.

Diagram showing the apothem of a regular hexagon

Definition

In a regular polygon, all sides and angles are equal. The apothem is the shortest distance from the center to any of its sides. It is an important concept in geometry because it helps in calculating the area of the polygon. The apothem can be used in the formula for the area of a regular polygon:

\( \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \)

Calculation

The apothem can be calculated if the side length and the number of sides of the regular polygon are known. For a regular polygon with side length \( s \) and \( n \) sides, the apothem \( a \) can be calculated using the formula:

\( a = \frac{s}{2 \tan(\pi/n)} \)

This formula arises from the fact that the apothem is the height of an isosceles triangle formed by two radii of the polygon and one of its sides.

Properties

• The apothem is always perpendicular to the side of the polygon.
• In a regular polygon, the apothem is the radius of the inscribed circle (incircle).
• The apothem is shorter than the radius of the circumscribed circle (circumcircle) of the polygon.

Applications

The apothem is used in various applications, including:

• Calculating the area of regular polygons.
• Determining the radius of the incircle of a regular polygon.
• In architectural design and engineering, where regular polygons are used in tiling and other structures.

Related pages

• Regular polygon
• Incircle and excircles of a triangle
• Circumscribed circle
• Tangent (trigonometry)