Editing Paraboloid (section)

== Dimensions of a paraboloidal dish ==
The dimensions of a symmetrical paraboloidal dish are related by the equation
<math display="block">4FD = R^2,</math>
where {{math|''F''}} is the focal length, {{math|''D''}} is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and {{math|''R''}} is the radius of the rim. They must all be in the same [[unit of length]]. If two of these three lengths are known, this equation can be used to calculate the third.

A more complex calculation is needed to find the diameter of the dish ''measured along its surface''. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: {{math|''P'' {{=}} 2''F''}} (or the equivalent: {{math|''P'' {{=}} {{sfrac|''R''{{sup|2}}|2''D''}}}}) and {{math|''Q'' {{=}} {{sqrt|''P''{{sup|2}} + ''R''{{sup|2}}}}}}, where {{math|''F''}}, {{math|''D''}}, and {{math|''R''}} are defined as above. The diameter of the dish, measured along the surface, is then given by
<math display="block">\frac{RQ}{P} + P \ln\left(\frac{R+Q}{P}\right),</math>
where {{math|ln ''x''}} means the [[natural logarithm]] of {{math|''x''}}, i.e. its logarithm to base {{math|''[[e (mathematical constant)|e]]''}}.

The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal [[wok]]), is given by
<math display="block">\frac{\pi}{2} R^2 D,</math>
where the symbols are defined as above. This can be compared with the formulae for the volumes of a [[Cylinder (geometry)|cylinder]] ({{math|π''R''{{sup|2}}''D''}}), a [[sphere|hemisphere]] ({{math|{{sfrac|2π|3}}''R''{{sup|2}}''D''}}, where {{math|''D'' {{=}} ''R''}}), and a [[Cone (geometry)|cone]] ({{math|{{sfrac|π|3}}''R''{{sup|2}}''D''}}). {{math|π''R''{{sup|2}}}} is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept. The surface area of a parabolic dish can be found using the area formula for a [[surface of revolution#Area formula|surface of revolution]] which gives
<math display="block">A = \frac{\pi R\left(\sqrt{(R^2+4D^2)^3}-R^3\right)}{6D^2}.</math>