Just as the ratio of the arc length of a semicircle to its radius
is always ,
the ratio
of the arc length of the parabolic segment formed
by the latus rectum of any parabola
to its semilatus rectum (and focal
parameter ) is a universal constant

(OEIS A103710 ). This can be seen from the equation of the arc length of a parabolic
segment

(5)

by taking
and plugging in and .

The other conic sections , namely the ellipse and hyperbola , do not have such universal constants
because the analogous ratios for them depend on their eccentricities .
In other words, all circles are similar and all parabolas are similar, but the same
is not true for ellipses or hyperbolas (Ogilvy 1990, p. 84).

The area of the surface generated by revolving for about the -axis is given by

(Love 1950, p. 288; OEIS A103713 ) and the area of the surface generated by revolving for about the -axis is

(Love 1950, p. 288; OEIS A103714 ).

The expected distance from a randomly selected point in the unit square to its center
(square point picking ) is

(Finch 2003, p. 479; OEIS A103712 ).

is an irrational number . It is also a transcendental
number , as can be seen as follows. If were algebraic, then would also be algebraic. But then, by
the Lindemann-Weierstrass theorem ,
would be transcendental, which is a contradiction.

The mean cylindrical radius of a hemicube constructed from unit cube is equal to .

See also Focal Parameter ,

Latus Rectum ,

Lindemann-Weierstrass Theorem ,

Parabola ,

Parabolic
Segment ,

Semilatus Rectum
This entry contributed by Sylvester
Reese

This entry contributed by Jonathan
Sondow

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References Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 479, 2003. Love,
C. E. Differential
and Integral Calculus, 4th ed. New York: Macmillan, 1950. Ogilvy,
C. S. Excursions
in Geometry. New York: Dover, 1990. Sloane, N. J. A.
Sequences A103710 , A103711 ,
A103712 , A103713 ,
and A103714 in "The On-Line Encyclopedia
of Integer Sequences." Referenced on Wolfram|Alpha Universal Parabolic Constant
Cite this as:
Reese, Sylvester and Sondow, Jonathan . "Universal Parabolic Constant." From MathWorld --A
Wolfram Web Resource, created by Eric W. Weisstein .
https://mathworld.wolfram.com/UniversalParabolicConstant.html

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