Paraboloid Volume Calculator

How it is calculated

The solid is a paraboloid of revolution: a parabola turned around its axis, with a flat circular opening of radius r and a depth (height) h measured along the axis from the tip to the plane of that opening. Its volume is V = ½ · π · r² · h. The tool also prints the volume of the enclosing cylinder, π · r² · h, on a line worded "½ of cylinder": the number on that line is the full cylinder, and the paraboloid is half of it. Your radius and height are echoed to four decimals.

No units are chosen: whatever unit you type for r and h, the result is in that unit cubed, although the output label reads "units³". Both values must be above zero or no result appears. The formula applies to a full, smooth paraboloid; it does not handle a truncated bowl (a frustum with the tip cut off), a dish with a rim, or a shell with wall thickness.

Common mistakes

• Entering the diameter in the radius field. The radius is half the opening width, so a 6 cm wide bowl has r = 3; entering 6 gives four times the volume.
• Using the slanted length along the surface as the height. The height must be measured straight down the axis, from the tip to the plane of the rim.
• Mixing units, for example radius in centimetres and height in metres. Both must use the same unit before the result is meaningful.
• Reading the answer as litres. Cubic centimetres divided by 1,000 give litres; 56.5487 cm³ is only about 0.057 litres.
• Applying the formula to a cone or a hemisphere. A cone with the same r and h holds one third of the cylinder and a paraboloid holds one half, so the shapes are not interchangeable.