Parabolic Segment Calculator

Name: Parabolic Segment - Geometry Calculator
Author: Jürgen Kummer

Calculations at a right parabolic segment. This is defined by a parabola of the form y=sx² in the interval x ∈ [ -a ; a ].
Enter the shape parameter s (s>0, normal parabola s=1) and the maximal input value a (equivalent to half the span) and choose the number of decimal places. Then click Calculate.

The parabola shape can be drawn with the function graph plotter.

Formulas:

h = s \cdot a^{2}

l = a \cdot \sqrt{1 + 4 s^{2} a^{2}} + \frac{\ln (2 s a + \sqrt{1 + 4 s^{2} a^{2}})}{2 s}

p = l + 2 a

A = \frac{4}{3} \cdot s \cdot a^{3}

ln is the logarithmus naturalis (natural logarithm).

Shape parameter, input value a, height, parabola arc length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
Although for the calculation of the area the value of the radius is taken to the power of 3, the unit is to the power of 2. This is, because the parabola function squares the length, but not the unit.

The formula for calculating the area is as follows: the rectangle in which the parabolic segment lies has an area of 2*a*h, i.e. 2*s*a³. The integral of s*a², i.e. s/3*a³, is subtracted from this twice in order to remove the areas to the left and right of the parabola. So 2*s*a³ - 2/3*s*a³ = 4/3*s*a³
The length of the parabolic arc is also calculated using integrals, but this is a lot more complicated.

A parabolic segment is created as a cutting surface when a cone is cut diagonally across the middle to the base. Other conic sections are circles, ellipses and hyperbolas. If you cut vertically you get an isosceles triangle.
The leading late antique Greek mathematician Hypatia of Alexandria is known to have worked on conic sections. She wrote at least a commentary on the works of Apollonius of Perga. However, the pagan Neoplatonist was murdered by a Christian mob in 415 or 416, and her writings were subsequently lost.

Calculation of area and other dimensions: Parabolic Segment

Last updated on 03/31/2026. Author: Jürgen Kummer

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