Apothem

From Wikipedia, the free encyclopedia

Apothem of a hexagon
Apothem of a hexagon

An apothem of a geometric figure is the perpendicular line segment, usually from the center, to an edge, or the length thereof.

The apothem of a regular polygon is a line drawn from the center of the polygon perpendicular to one of its sides. (In other words, a line drawn from the center to the midpoint of the side.)

The apothem can be used to find the area of regular polygons, the area equalling half of the perimeter multiplied by the length of the apothem:

A = {p a \over 2}

The apothem is also the radius of the inscribed circle. Given a regular n-polygon with apothem r, radius of the circumscribed circle R and length of one of the sides z one has:

r=\frac{z}{2\tan(\pi/n)}=R\cos(\pi/n)

For a given circle and chord, the apothem is the distance from the midpoint of the chord to the centre of the circle.

For a pyramid (geometry), an apothem is the height of a lateral face, that is, a shortest distance from apex to base, the slant height of the pyramid; for a truncated pyramid, the height of a trapezoidal lateral face is an apothem.[1]

Depending upon what information is provided, there are multiple ways to find an apothem.

  • Side length 'x' and number of sides 'n'.
Apothem=\frac{1}{2}x(tan(\frac{90(n-2)}{n}))
  • Perimeter

The perimeter is basically the same; however, 'x' side length would be replaced with perimeter 'p' and number of sides 'n':

\frac{p}{n}

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