Knoll Calculator

Name: Knoll - Geometry Calculator
Author: Jürgen Kummer

Calculations at a knoll. Here, a knoll is an isosceles trapezoid with a matching circular segment attached in a way, so that at the connection between trapezoid and segment no vertex will appear.
Enter the three different trapezoid lengths, choose the number of decimal places and click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units. The angle of the circular segment is twice the angle α, Θ = 2α.

Formulas:

j = \frac{1}{2} \cdot \sqrt{4 c^{2} - {(a - b)}^{2}}

α = \arccos [\frac{{(\frac{a - b}{2})}^{2} + c^{2} - j^{2}}{(a - b) \cdot c}]

i = \sqrt{\frac{\frac{b^{2}}{4}}{\frac{2}{[1 - \cos (α)]} - 1}}

h = i + j

l = i \cdot \frac{2 α}{1 - \cos (α)}

p = a + 2 c + l

A = \frac{\sqrt{{(a + b)}^{2} \cdot (a - b + 2 c) \cdot (b - a + 2 c)}}{4} + \frac{l^{2}}{4 α} - \frac{b \cdot (\frac{l}{2 α} - i)}{2}

(α in rad)

Lengths, heights and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The three-dimensional equivalent of such a knoll is the rounded cone, which is formed when a knoll is rotated about its perpendicular bisector. Analogically, a knoll can also be described as a rounded isosceles triangle. The perpendicular bisector is the line perpendicular through the center of the base, along the line of the altitude i of the circular segment. Like the isosceles triangle, the knoll is axially symmetric about this perpendicular bisector.

The challenge in this calculation is finding the perfectly fitting circular segment for the isosceles trapezoid. To do this, the angle α is derived from the trapezoid's dimensions using the Side-Angle-Side (SAS) theorem for triangles. This is the most complex step. From this angle and the segment's chord length, which must be equal to the shorter side of the trapezoid, the circle's radius, and thus the segment's other dimensions, can be calculated.

This knoll is an example of how to round off a single corner. To round off any corner with an angle β, β must be 180 degrees minus two α, i.e., α = (180° - β)/2. Using this α and the width of the rounding, the required circular segment can then be determined.
For other two-dimensional shapes with multiple rounded corners, see rounded polygon and rounded rectangle, whereas for those all corners are equal. For different corners, the calculation becomes correspondingly more complex.

Calculation of area and other dimensions: knoll

Last updated on 04/09/2026. Author: Jürgen Kummer

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