What does apothem mean?
An apothem is the perpendicular distance from the centre of a regular polygon to the midpoint of any of its sides. Because all sides of a regular polygon are equidistant from the centre, this single length characterises the entire figure, and it equals the radius of the circle inscribed within the polygon. The apothem is central to the standard formula for the area of a regular polygon: the area equals half the perimeter multiplied by the apothem, a relationship analogous to the triangle area formula of half base times height. For example, a regular hexagon's area can be computed directly once its apothem is known. The word derives from Greek roots meaning 'set away' or 'deposited', reflecting the idea of a measured distance laid off from the centre. A technical term confined largely to mathematics education and geometric proof, the apothem is nonetheless indispensable for anyone working with polygons, tiling, or symmetry, offering an elegant bridge between a shape's perimeter and its enclosed area.
nounThe perpendicular distance from the centre of a regular polygon to any one of its sides; equivalently, the radius of the polygon's inscribed circle.
- The perpendicular distance from the centre of a regular polygon to one of its sides, equal to the radius of its inscribed circle.
"The area of a regular hexagon can be found by multiplying half its perimeter by its apothem."
"To find the area of a regular pentagon, multiply half the perimeter by the apothem."
"The apothem of a unit square is exactly one-half, since it measures the distance from the centre to each side."
Rarely pluralised in practice, since a given regular polygon has exactly one apothem; the plural appears mainly when comparing several different polygons.
"As the number of sides increases, the apothems of the successive polygons approach the radius of the circumscribing circle."
Every regular polygon hides a secret measurement called the apothem — and without it, the classic area formula for shapes like hexagons simply wouldn't work.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of apothem
The term 'apothem' comes from the Greek 'apothema' (ἀπόθεμα), formed from 'apo-', meaning 'away from' or 'off', combined with 'thema', meaning 'that which is laid down or set'. It thus literally conveys a distance laid off from the centre toward a side. The word entered European mathematical vocabulary through Latinised Greek learning, becoming established in English geometrical texts by around the early modern period. It shares its root element with related Greek-derived terms built on the notion of 'setting' or 'placing', such as 'theme' and 'thesis'.
How apothem is actually used
Strictly a technical term of geometry, used almost exclusively in mathematical contexts such as textbooks, proofs, and formula sheets. It applies only to regular polygons — figures with all sides and angles equal — and is rarely encountered outside that setting.
Easily confused with apothem
In a regular polygon, the radius runs from the centre to a vertex, whereas the apothem runs perpendicularly from the centre to the midpoint of a side.
Despite the shared 'apo-' beginning, apotheosis means elevation to divine status and has nothing to do with geometry.