The apothem (sometimes abbreviated as apo) of a

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

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 that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

that is

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...

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

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

''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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

. For a regular

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

, which is a pyramid whose base is a regular polygon, the apothem is the

slant height Slant can refer to: Bias *Bias or other non-objectivity in journalism, politics, academia or other fields Technical * Slant range, in telecommunications, the line-of-sight distance between two points which are not at the same level * Slant d ...

of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a plane parallel to the base), the apothem is the height of a

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...

al lateral face. For an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

, the apothem is equivalent to the line segment from the midpoint of a side to the triangle's center.

Properties of apothems

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

area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 ope ...

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 a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...

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

A = \frac = \frac.

This formula can be derived by partitioning the ''n''-sided polygon into ''n''

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) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of the inscribed circle. 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 organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

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

Properties of apothems

Finding the apothem

Notes

See also

References

External links