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Calculate volume, surface area, and focal length of a paraboloid of revolution. Used for satellite dishes and reflectors.
A paraboloid of revolution is created by rotating a parabola around its axis of symmetry. Its key property is that parallel rays hitting the surface all reflect through the focal point.
This reflective property makes paraboloids ideal for satellite dishes, radio telescopes, flashlight reflectors, and solar concentrators. The receiver is placed at the focus.
Satellite dishes, radio telescopes, car headlight reflectors, flashlight reflectors, solar cookers, parabolic microphones, and some architectural roofs.
Place the receiver at the focal point, which is f = r²/(4h) above the vertex. For a dish with r = 1m and h = 0.25m, the focus is at f = 1/(4×0.25) = 1m from the bottom.
Using calculus (integration of circular cross-sections), the paraboloid's volume comes out to exactly half the cylinder that would contain it.
When the cross-sections are ellipses instead of circles. It has two different focal lengths. The formulas here assume circular cross-sections (paraboloid of revolution).