Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. The perpendicular distance from the center of a regular polygon to any of its sides.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The perpendicular distance from the center of a circle to a chord of the same circle.
 n. The distance from the center of a regular polygon perpendicular to one of its sides (a special case of the above).
from the GNU version of the Collaborative International Dictionary of English
 n. The perpendicular from the center to one of the sides of a regular polygon.
 n. A deposit formed in a liquid extract of a vegetable substance by exposure to the air.
from The Century Dictionary and Cyclopedia
 n. In geometry, a perpendicular let fall from the center of a regular polygon upon one of its sides.
 n. In pharmaceutics, the more or less completely insoluble brownish substance deposited when vegetable infusions, decoctions, tinctures, etc., are subjected to prolonged evaporation by heat with access of air. The substance or substances out of which it is in this way formed constitute the socalled extractive.
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

He previously gained points to qualify for the semifinal round by correctly spelling "apothem" and "caponata."

When gm is final drop the silent g: apothem, diagram, flem.

Further, as M. M.urice Girard has pointed out, the apothem of the cell varies among different races of bees, so that the standard would alter from hive to hive, according to the species of bee that inhabited it.

How do you find the area of an equilateral traingle with a radius of 12 and an apothem of 6?

The isoceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semibase by apothem), joining the mediumlength edges to make the apothem.

• A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semibase (b), and height (h); the face inclination angle is also marked.

Mathematical pyramids • A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semibase (half the base width) is sometimes called a golden pyramid.

The slant height or apothem is 5 / 3 or 1.666 ... times the semibase.

· new shapes: regular polygon by center, vertex or apothem, and vertex number

a clear infusion of such leaves, evaporated carefully to dryness, was not all undissolved by water, but left a quantity of brown oxidised extractive matter, to which the denomination _apothem_ has been applied by some chemists; a similar result is obtained by the evaporation of an infusion of black tea.
Chambers's Edinburgh Journal, No. 442 Volume 17, New Series, June 19, 1852
hernesheir commented on the word apothem
(n): In geometry, a perpendicular from the center of a regular polygon to a side. NOT to be confused with apothegm.
January 15, 2009