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Contradiction

"Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent and have the same length."
"A triangle has four centers, circumcenter, incenter, centroid, and orthocenter. The center that is used to find the apothem is the incenter."

1 renders 2 a null statement, since for an equilateral triangle all centres coincide. I can therefore make out only that either:

whoever wrote 1 was considering a triangle not to count as a polygon, which contradicts the view taken at Polygon, and that the definition of apothem treats triangles specially
somebody just didn't know what he/she was talking about

What is correct? ISTM the real definition of apothem should be a generalisation of both statements: for any polygon that has an inscribed circle (to which every side is a tangent), a radius of this circle that meets a side of the polygon (and is therefore perpendicular to it). But can anybody find a good source on the term? (MathWorld contradicts itself too - it defines the apothem of a general regular polygon as the same as the inradius, which it defines only for general triangles and polyhedra.)

On the basis of this definition, all apothems of a given polygon are the same length not for the reason given in 1, but because they are radii of the circle. What was the writer of 1 (mis)using congruent to mean, anyway? -- Smjg (talk) 11:46, 12 February 2009 (UTC)[reply]

Mathmatical symbols used

Perhaps links to other Wikipedia articles for the first time mathematical symbols, constants and formulae are used in the formulas? I know what cos(ine), tan(gent) and Pi are, but other people might not know what these things are. You have to start somewhere with Maths.

Some people might not know that Pi and the symbol for Pi are the same thing.

I don't know what the . dot is at the end of some of those formulas, Is it a decimal point, a full stop (AKA a period) or something else? I think it would be helpful if that symbol had an explanation, as it appears at the end of some of the formulae and not others.

StuZealand (talk) 12:50, 17 March 2021 (UTC)[reply]

Contradiction (2)

1. 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.

2. 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.

Is 2 a definition specifically for regular pyramids, separate from 1? If not, then the only expected appearances of apothems in a regular pyramid are in its base, it being a regular polygon. Unless coincidentally being of the same length, the apothem is not the slant height of a lateral face; rather, only in a right regular pyramid, does the apothem form a right-angled triangle with the aforesaid slant height and the perpendicular dropped from the apex to the base of the pyramid (the height of the pyramid). The mention of any relationship between apothems and the heights of faces of truncated regular pyramids is even more confusing.

Sfa00062 (talk) 16:48, 11 April 2021 (UTC)[reply]

@Sfa00062: I agree with you and since that paragraph on pyramids is unsourced, I'll remove it. Powers ^T 12:35, 23 June 2023 (UTC)[reply]

confusion on formulas for apothem?

It should be made clear that when taking the tangent in the formulas, you doing so using radiants, not degrees. — Preceding unsigned comment added by Warblerab295 (talk • contribs)