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A more complex calculation is needed to find the diameter of the dish ''measured along its surface''. This is sometimes called the "linear diameter". It is 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>\scriptstyle P=2F</math> (or equivalently <math>\scriptstyle P=\frac{R^2}{2D})</math> and <math>\scriptstyle Q=\sqrt {P^2+R^2},</math> where <math>\scriptstyle F,</math> <math>\scriptstyle R,</math> and <math>\scriptstyle D</math> are defined as above. The linear diameter, <math>\scriptstyle L,</math> is then given by: <math>\scriptstyle L= \frac {RQ} {P} + P \ln \left ( \frac {R+Q} {P} \right ),</math> where <math>\scriptstyle \ln(x)</math> means the [http://en.wikipedia.org/wiki/Natural_logarithm natural logarithm] of <math> \scriptstyle x </math>, i.e. its logarithm to base "e". (This calculation is exactly accurate. Computer programs that approximate a parabola with a short sequence of straight line segments are less accurate. This includes some published programs.) | A more complex calculation is needed to find the diameter of the dish ''measured along its surface''. This is sometimes called the "linear diameter". It is 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>\scriptstyle P=2F</math> (or equivalently <math>\scriptstyle P=\frac{R^2}{2D})</math> and <math>\scriptstyle Q=\sqrt {P^2+R^2},</math> where <math>\scriptstyle F,</math> <math>\scriptstyle R,</math> and <math>\scriptstyle D</math> are defined as above. The linear diameter, <math>\scriptstyle L,</math> is then given by: <math>\scriptstyle L= \frac {RQ} {P} + P \ln \left ( \frac {R+Q} {P} \right ),</math> where <math>\scriptstyle \ln(x)</math> means the [http://en.wikipedia.org/wiki/Natural_logarithm natural logarithm] of <math> \scriptstyle x </math>, i.e. its logarithm to base "e". (This calculation is exactly accurate. Computer programs that approximate a parabola with a short sequence of straight line segments are less accurate. This includes some published programs.) | ||
The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom, is given by <math>\scriptstyle V= \frac {1} {8 F}\pi R^4 ,</math> or equivalently <math>\scriptstyle V=2\pi F D^2,</math> or <math>\scriptstyle V= \frac {1} {2} \pi R^2 D ,</math> where the symbols are defined as above. This last version can be compared with the well-known formulae for the volumes of a cylinder and a cone. Of course, <math>\scriptstyle \pi R^2 </math> is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight the reflector dish can intercept. | The volume of the dish, i.e. the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom, is given by <math>\scriptstyle V= \frac {1} {8 F}\pi R^4 ,</math> or equivalently <math>\scriptstyle V=2\pi F D^2,</math> or <math>\scriptstyle V= \frac {1} {2} \pi R^2 D ,</math> where the symbols are defined as above. This last version can be compared with the well-known formulae for the volumes of a cylinder and a cone. Of course, <math>\scriptstyle \pi R^2 </math> is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight the reflector dish can intercept. | ||
The surface area of a paraboloidal dish can be found using the area formula for a surface of revolution, which gives <math>\scriptstyle A=\frac{\pi R}{6 D^2}\left((R^2+4D^2)^{3/2}-R^3\right)</math>, providing <math>\scriptstyle D \ne 0</math>. | The surface area of a paraboloidal dish can be found using the area formula for a surface of revolution, which gives <math>\scriptstyle A=\frac{\pi R}{6 D^2}\left((R^2+4D^2)^{3/2}-R^3\right)</math>, providing <math>\scriptstyle D \ne 0</math>. |