Sphere Difference Calculator

Name: Sphere Difference - Geometry Calculator
Author: Jürgen Kummer

Calculations for a sphere difference. A sphere difference occurs when one sphere is subtracted from two intersecting spheres. The two spheres do not need to be the same size for this calculator. Two spherical caps are removed from the remaining sphere.
Enter the radii of the two spheres and the distance between their centers. Choose the number of decimal places, then click Calculate. The distance must be less than the sum of the two radii but greater than their difference.

Formulas:

x = \frac{a^{2} - b^{2} + c^{2}}{2 c}

r = \sqrt{a^{2} - x^{2}}

h_{A} = a - x

h_{B} = b - (c - x)

d = c - b

A = 4 π a^{2} - 2 π a h_{A} + 2 π b h_{B}

V = \frac{4 \cdot π \cdot a^{3}}{3} - \frac{π \cdot {h_{A}}^{2}}{3} \cdot (3 a - h_{A}) - \frac{π \cdot {h_{B}}^{2}}{3} \cdot (3 b - h_{B})

Kreiszahl pi:

π = 3.141592653589793...

Radius, distances and heights 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.

This shape is particularly well-known from the Death Star in the Star Wars films. Saturn's moon Mimas, which is also nicknamed the Death Star, also has simplified this shape.

When the two spheres intersect, a shape is formed from two spherical caps, like a lens without the central part. The two caps have the same base area but different heights. The radius of this circular base is r, and its center is located at a distance x from the center of the remaining sphere. The outer cap is the removed spherical cap of the remaining sphere, and the trough is the spherical cap of the removed sphere. The distance d from the center of the remaining sphere to the lowest point of the trough can be negative, in which case the trough extends beyond this center point. The area is calculated as the area of the first sphere minus the area of the calotte of the removed spherical cap of the remaining sphere, plus the area of the calotte of the trough. The volume is that of the first sphere minus the volumes of both spherical caps.

The geometric shape of the sphere difference is a solid of revolution. It is rotationally symmetric about the axis of rotation passing through the center of the remaining sphere and the lowest point of the trough. It is mirror-symmetric about any plane in which this axis lies.

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