apothem

The term apothema ( ancient Greek ἀπόθεμα , deposit ' ) is a historical name in geometry for the perpendicular from the center of a circle to a chord of the circle. The length of the apothema is thus the distance between the chord and the center of the circle and is equal to the difference between the radius of the circle and the height of the segment of the circle , which is historically known as sagitta . In a regular polygon that is inscribed in a circle, all apothems are congruent to one another and their lengths correspond to the inscribed radius of the polygon.

definition

Chord AB of a circle k with radius r, chord length l, sagitta h and apothema a

If a circle with a center and a chord of the circle with a center , then the apothema of the chord is defined as ${\displaystyle k}$ ${\displaystyle M}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle L}$ ${\displaystyle a}$

${\displaystyle a:={\overline {ML}}}$,

i.e. the length of the connecting line between the center of the circle and the center of the chord. The apothema is thus the plumb line from the center of the circle to the tendon, where the plumb line is the base. In addition to the naming , the distance between the tendon and the center of the circle is also called the apothema. ${\displaystyle L}$ ${\displaystyle a}$ ${\displaystyle {\overline {ML,}}}$

calculation

If the radius of the circle is the length of the chord, then, according to the Pythagorean theorem, the length of the apothemas applies ${\displaystyle r}$ ${\displaystyle l}$ ${\displaystyle a}$

${\displaystyle r^{2}=a^{2}+\left({\frac {l}{2}}\right)^{2}}$

and thus

${\displaystyle a={\sqrt {r^{2}-{\tfrac {l^{2}}{4}}}}}$.

In a regular polygon with sides and the side length , the length of each apothemas corresponds to the inscribed radius of the polygon and thus ${\displaystyle n}$ ${\displaystyle l}$

${\displaystyle a={\frac {l}{2\,\tan \left({\frac {180^{\circ }}{n}}\right)}}}$.

With the aid of the apothema, the area of a regular polygon can also be determined as, since a regular polygon can be broken down into congruent isosceles triangles of the area . The following values ​​result for various regular polygons: ${\displaystyle A={\tfrac {1}{2}}nla}$ ${\displaystyle n}$ ${\displaystyle {\tfrac {1}{2}}la}$

Apothema of a regular hexagon

regular polygon	Side length	apothem	surface
triangle	${\displaystyle l=r\cdot {\sqrt {3}}}$	${\displaystyle a=r\cdot {\tfrac {1}{2}}}$	${\displaystyle A=r^{2}\cdot {\tfrac {3{\sqrt {3}}}{4}}}$
square	${\displaystyle l=r\cdot {\sqrt {2}}}$	${\displaystyle a=r\cdot {\tfrac {\sqrt {2}}{2}}}$	${\displaystyle A=r^{2}\cdot 2}$
pentagon	${\displaystyle l=r\cdot {\sqrt {{\tfrac {1}{2}}(5-{\sqrt {5}})}}}$	${\displaystyle a=r\cdot {\tfrac {1}{4}}(1+{\sqrt {5}})}$	${\displaystyle A=r^{2}\cdot {\tfrac {5}{8}}{\sqrt {(10+2{\sqrt {5}})}}}$
hexagon	${\displaystyle l=r\,}$	${\displaystyle a=r\cdot {\tfrac {1}{2}}{\sqrt {3}}}$	${\displaystyle A=r^{2}\cdot {\tfrac {3}{2}}{\sqrt {3}}}$
octagon	${\displaystyle l=r\cdot {\sqrt {2-{\sqrt {2}}}}}$	${\displaystyle a=r\cdot {\tfrac {1}{2}}{\sqrt {2}}}$	${\displaystyle A=r^{2}\cdot 2{\sqrt {2}}}$
${\displaystyle n}$Corner	${\displaystyle l=r\cdot 2\cdot \sin {\tfrac {180^{\circ }}{n}}}$	${\displaystyle a=r\cdot \cos {\tfrac {180^{\circ }}{n}}}$	${\displaystyle A=r^{2}\cdot {\tfrac {n}{2}}\cdot \sin {\tfrac {360^{\circ }}{n}}}$
${\displaystyle n\to \infty }$ (Circle)	${\displaystyle l\to 0}$	${\displaystyle a\to r}$	${\displaystyle A\to r^{2}\cdot \pi }$

See also

• Sagitta method

literature

• Köberlein, J. Michael: Textbook of elementary geometry and trigonometry initially for high schools and high schools , Sulzbach 1824

Web links

Commons : Chord (geometry)  - collection of images, videos and audio files

Wiktionary: Apothema  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Apothem . In: MathWorld (English).
• Sagitta, Apothem, and Chord Ed Pegg, Jr., The Wolfram Demonstrations Project

Individual evidence

1. ^ Paul Huther: The beginnings of geometry, excellent for use in technical schools. , Regensburg, 1838. Published by G. Joseph Manz. Sixth section. Of the figures in and around the circle, §79. Addendum 3., p. 42 (last sentence) online copy (Google), accessed on January 16, 2019.