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How to Calculate a Solid Angle Parabola

We usually think of a parabola as being two-dimensional, a line on a plane. However, imagine that plane rotating about the axis of the parabola. You would then have a three-dimensional figure, a solid, defined by the sweep of the parabola through space. If you cut off that figure at some distance "h" from the vertex of the parabola, you have a solid that is shaped sort of like an igloo without an entrance, much like a rounded cone. Technically, this is a "paraboloid of revolution." It is useful to know how to calculate the volume of this solid.

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Instructions

- 1
  Determine the height of the paraboloid. Imagine the parabola placed on your desk like an igloo-shaped paperweight. The height, "h," is the direct distance from the apex of the parabola, or the top of the igloo, to the desk.
- 2
  Determine the radius of the paraboloid. Imagine a line dropping directly down from the apex of the parabola to a point, "p," on the desk, centered directly underneath the apex. The radius, "R," is the distance from "p" to the edge of the parabola, as measured along the desk.
- 3
  Calculate the volume of the paraboloid. The volume, "V," is calculated as follows: V = (0.5) (pi) (R squared) (h) . So, if "R" equals 2 cm, and "h" equals 3 cm, then V = (0.5) (pi) (2 squared) (3) = (0.5) (3.14) (4) (3) = 18.84 cubic cm.

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