Question

Area of the shaded portion of a cylinder

Answer 1

Definition: A shape formed when a cylinder is cut by a plane parallel to the sides of the cylinder.
Try this Drag the orange dots, note how the volume changes.See also: Volume of a cylinder

If we take a horizontal cylinder, and cut it into two pieces using a cut parallel to the sides of the cylinder, we get two horizontal cylinder segments. In the figure above, the bottom one is shown colored blue. The other one is the transparent part on top.

If we look a the end of the cylinder, we see it is a circle cut into two circle segments. See Circle segment definition for more.

Whenever we have a solid whose cross-section is the same along its length, we can always find its volume by multiplying the area of the end by its length. So in this case, the volume of the cylinder segment is the area of the circle segment, times the length.

So as a formula the volume of a horizontal cylindrical segment is

volume=slWhere 
   s = the area of the circle segment forming the end of the solid, and
   l = the length of the cylinder.

The area of the circle segment can be found using it's height and the radius of the circle. 
See Area of a circle segment given height and radius.

Calculator

Use the calculator below to calculate the volume of a horizontal cylinder segment. It has been set up for the practical case where you are trying to find the volume of liquid is a cylindrical tank by measuring the depth of the liquid.

For convenience, it converts the volume into liquid measures like gallons and liters if you select the desired units. If you do not specify units the volume will be in whatever units you used to input the dimensions. For example, if you used feet, then the volume will be in cubic feet. Use the same units for all three inputs.

Units    None    Metric    US                 Cylinder diameterCylinder lengthDepthVolume Calculate Clear As a formulavolume=L R2cos−1 R−DR −(R−D)√2RD−D2 where:
R  is the radius of the cylinder.
D  is the depth.
L   is the length of the cylinder

Notes:The result of the cos-1 function in the formula is in radians.The formula uses the radius of the cylinder. This is half its diameter.All inputs must be in the same units. The result will be in those cubic units. So for example if the inputs are in inches, the result will be in cubic inches. If necessary the result must be converted to liquid volume units such as gallons.Related topicsDefinition of a faceDefinition of amn edgeVolumeDefinition and properties of a cubeVolume enclosed by a cubeSurface area of a cubeDefinition and properties of a pyramidOblique and right pyramidsVolume of a pyramidSurface area of a pyramidCylinder - definition and propertiesCylinder relation to a prismCylinder as the locus of a lineOblique cylindersVolume of a cylinderVolume of a partially filledcylinderSurface area of a cylinderPrism definitionVolume of a prismSurface area of a prismVolume of a sphereSurface area of a sphereDefinition of a coneOblique and Right ConesVolume of a coneSurface area of a coneDerivation of the cone area formulaSlant height of a coneConic sections - the circleConic sections - the ellipseIcosahedron (20 faces each an equilateral triangle)(C) 2011 Copyright Math Open Reference. All rights reserved