Hollow Cone Calculator

Name: Hollow Cone - Geometry Calculator
Author: Jürgen Kummer

Calculations at a hollow cone. This is a right circular cone, of which a smaller, similar cone is removed at the center of its base.
Enter one radius, one height and one further value of radius, height and thickness. Choose the number of decimal places, then click Calculate.

Formeln:

\frac{R}{r} = \frac{H}{h}

a = R - r

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

V = \frac{π}{3} \cdot (R^{2} \cdot H - r^{2} \cdot h)

pi:

π = 3.141592653589793 ...

Radiuses, heights and thickness have the same unit (e.g. meter), the surface has 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 original cone and the removed cone are similar. In geometry, similarity is precisely defined. Simply put, it means that two shapes possess the same proportions. They need not be of the same size. However, if they can be scaled to the same size via a homogeneous dilation, then rotated and, if necessary, reflected so that they fit together exactly, then these solids are termed similar. Similar solids share the same formulas for calculating their properties, only the values themselves depend on the specific size of the solid. The formulas presented above are therefore composed of two sets of cone formulas, one for the large outer cone and one for the small inner cone. The volume of the hollow cone is simply the volume of the outer cone minus that of the inner cone. Calculating the surface area is somewhat more complex. It consists of the surface area of the outer cone, minus the base area of the inner cone, plus the lateral surface area of the inner cone.

The hollow cone possesses the same symmetry properties as a solid cone. It is rotationally symmetric with respect to any rotation about the axis passing through both vertices and the centers of the base surfaces, and mirror-symmetric with respect to any plane containing this axis of rotation. The base of the hollow cone is an annulus, the cross-section through one plane of symmetry is an arrow-hexagon.

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