Rounded Cone Calculator

Name: Rounded Cone - Geometry Calculator
Author: Jürgen Kummer

Calculations at a rounded cone. This is a truncated cone with a matching spherical cap attached in a way, so that at the connection between cone and cap no edge will appear. The vertical central slice plane of a rounded cone is a knoll. For the designations of the variables in the sketch, see there. The horizontal slice planes are all circles.
Enter radiuses and height of the truncated cone, choose the number of decimal places and click Calculate. The angle is calculated and displayed in degrees, here you can convert angle units.

Formulas:

α = \arccos [\frac{(R - r)}{\sqrt{h^{2} + {(R - r)}^{2}}}]

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

h = i + j

s = \frac{r^{2} + i^{2}}{2 i}

A = (R + r) \cdot π \cdot \sqrt{{(R - r)}^{2} + j^{2}} + π R^{2} + π \cdot (r^{2} + i^{2})

V = \frac{π}{3} \cdot {j \cdot (R^{2} + R r + r^{2}) + i^{2} \cdot [\frac{3 \cdot (r^{2} + i^{2})}{2 i} - i]}

pi:

π = 3.141592653589793 ...

Radiuses and heights height have the same unit (e.g. meter), surfaces have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

The rounded cone is a solid of revolution. It is rotationally symmetric when rotated about the axis through the center of the calotte (the curved part of the spherical segment) and the center of the cone's base. It is also mirror-symmetric about any plane in which this axis lies. This shape consists of three surfaces: a spherical calotte, a lateral surface of a circular cone, and a circle. The calotte and the lateral surface merge seamlessly, a rounded edge is separating the lateral surface and the circle. The volume of the rounded cone is of course smaller than that of the original cone, it is comprised of the volume of the truncated cone and the volume of the spherical segment.

A circular cone is a tip with a round cross-section. Rounding off points is a common method used to make objects safer, thus reducing the risk of injury and damage. Rounding based on a sphere is mathematically the simplest, but not the only method. The larger the radius of the sphere of which the segment is taken, the blunter is the tip and the more was removed from the original cone.

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