The dimensional components of a three-dimensional solid are height, width and length. The volume of a solid is the amount of three-dimensional space that it occupies, which may be calculated from these lineal dimensions. The volume of some simple solids may be calculated arithmetically from their dimensions, while more complicated shapes require integral calculus to calculate their volume. Practical applications require volume to be expressed in units of cubed linear measure, such as cubic inches. However, purely theoretical calculations typically ignore units of measure.
Calculate the volume of a rectangular prism. This type of solid has six rectangular faces, and its volume is given as V = lwh, where V is the volume and l, w and h represent the linear dimensions of the solid.
Calculate the volume of a cylinder. We will use the radius r as the first of two dimensions to find the area of the cylinder's base and then multiply by the height h for the third dimension. The base is a circle, so its area is ?r^2, and the volume of a cylinder is therefore ?hr^2.
Find the volume of a pyramid from its linear dimensions. Use the length and width to find the area of the base and multiply the area by 1/3h. For a square pyramid with a base of length a, we have a^2 as the area of the base, so its volume would be (a^2)h/3.
Find the volume of a sphere from its dimension. From integral calculus, we have V = 4/3 ?r^3. Note that we use the radius as all three linear dimensions to calculate the volume.
Use integral calculus to find the volume of more complicated solids. To get the volume of a solid, we integrate the function A(h) with respect to h where A(h) is a function that provides the area of the cross-section at height h. This will work for any solid so long as A(h) is integrable for all values of h.