Home >> Science & Nature >> Science

How to Calculate the Surface Area of a Paraboloidal Object

Paraboloids are special forms of parabolas formed by rotating a parabola about its axes. The two types of paraboloids are elliptical and hyperbolic. Elliptical paraboloids are derived from a parabola rotated about its vertical y-axis; a hyperbolic paraboloid is derived from a parabola rotated about its z-axis, the axis pointing "out" of the page. The surface area of a paraboloid is described by a single equation solvable when you know the height and radius of a paraboloid.

Things You'll Need

Calculator

Instructions

- 1
  Add the squared value of the radius "a" of the paraboloid to the squared value of the height of the paraboloid "h" multiplied by 4. The symbolic expression looks like this: (a² + 4h²).
- 2
  Raise (a² + 4h²) by three halves to result in the expression (a² + 4h²)^(3/2).
- 3
  Subtract the cubed value of the radius from the result of the previous expression, giving: (a² + 4h²)^(3/2) - a³).
- 4
  Multiply the value of the previous expression by the term (?a) / 6h²) to result in the equation: Area = (?a) / 6h²) * (a² + 4h²)^(3/2) - a³). The result of the equation is the surface area of a paraboloidal object.

Science