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The Collaborative International Dictionary

Apothem \Ap"o*them\, n. [Gr. ? + ? that which is placed, ? to place.]

  1. (Math.) The perpendicular from the center to one of the sides of a regular polygon.

  2. A deposit formed in a liquid extract of a vegetable substance by exposure to the air.


n. 1 (context geometry English) The perpendicular distance from the center of a circle to a chord of the same circle. 2 (context geometry English) The distance from the center of a regular polygon perpendicular to one of its sides (a special case of the above).


The apothem (sometimes abbreviated as apo) of a regular polygon 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 that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent.

For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the slant height 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 trapezoidal lateral face.

For an equilateral triangle, the apothem is equivalent to the line segment from the midpoint of a side to any of the triangle's centers, since an equilateral triangle's centers coincide as a consequence of the definition.