The apothem (sometimes abbreviated as apo) of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

that is

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

to one of its sides. The word "apothem" can also refer to the length of that line segment and comes from the

ancient Greek Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...

''ἀπόθεμα'' ("put away, put aside"), made of ''ἀπό'' ("off, away") and ''θέμα'' ("that which is laid down"), indicating a generic line written down. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent.

Properties of apothems

The apothem ''a'' can be used to find the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of any regular ''n''-sided polygon of side length ''s'' according to the following formula, which also states that the area is equal to the apothem multiplied by half the

perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...

since ''ns'' = ''p''. :

A = \frac = \frac.

This formula can be derived by partitioning the ''n''-sided polygon into ''n'' congruent isosceles triangles, and then noting that the apothem is the height of each triangle, and that the area of a triangle equals half the base times the height. The following formulations are all equivalent: :

A = \tfracnsa = \tfracpa = \tfracns^2\cot\frac = na^2\tan\frac

An apothem of a regular polygon will always be a

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the

inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incente ...

. It is also the minimum distance between any side of the polygon and its center. This property can also be used to easily derive the formula for the area of a circle, because as the number of sides approaches infinity, the regular polygon's area approaches the area of the inscribed circle of radius ''r'' = ''a''. :

A = \frac = \frac = \pi r^2

Finding the apothem

The apothem of a regular polygon can be found multiple ways. The apothem ''a'' of a regular ''n''-sided polygon with side length ''s'', or circumradius ''R'', can be found using the following formula: :

a = \frac = R\cos\frac.

The apothem can also be found by :

a = \frac\tan\frac.

These formulae can still be used even if only the perimeter ''p'' and the number of sides ''n'' are known because ''s'' = .

Notes

References

External links

Apothem of a regular polygon
With interactive animation

* {{cite web , title=Sagitta, Apothem, and Chord , last=Pegg , first=Ed Jr., author-link=Ed Pegg, Jr. , publisher=

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...

, url=http://demonstrations.wolfram.com/SagittaApothemAndChord/ Polygons

Properties of apothems

Finding the apothem

Notes

See also

References

External links