Sphere Cone Calculator

Name: Sphere-Cone - Geometry Calculator
Author: Jürgen Kummer

Calculations for what is called here a sphere cone. This is a cone onto which a spherical cap is placed, having the same radius as the cone's base. The bases of the cone and the cap are therefore two congruent circles. A special case of such a sphere cone is the spherical sector, where the apex of the cone lies at the center of the sphere from which the spherical cap comes.
Enter two of the heights of the cone, the spherical cap and the sphere cone, as well as the radius of the circle at which they are joined. Choose the number of decimal places, then click Calculate.

Formulas:

h = i + j

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

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

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

pi:

π = 3.141592653589793...

Heights and radius have the same unit (e.g. meter), the area 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 spherical cap on the sphere cone is at most a hemisphere. Therefore, it generally doesn't work for this shape to be a cone containing a whole sphere, because the sphere's radius is usually larger than the radius of the cone's base. However, if the spherical cap is a hemisphere, meaning the sphere is halfway inside the cone, then it can work for very tall cones. The spherical cap is a hemisphere if j=r. If the cone's height is at least twice the radius, i≥2r, then the whole sphere fits inside without protruding beyond the cone's lateral surface.
Such a shape with seamless transition, where the spherical cap is larger than a hemisphere is the drop.

The surface area of such a sphere cone is calculated as the sum of the area of the calotte area of the spherical cap plus the lateral surface area of the cone. The volume of the sphere cone is the sum of the volumes of the two underlying figures. The radius of the sphere is obtained from the formula for the radius of the spherical cap, there solved for r.

The sphere cone is a solid of revolution. Its axis of rotation runs from the apex of the cone to the center of the curved surface, the cap. It is mirror-symmetric about every plane in which the axis of rotation lies.

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