Mensuration, the name of that branch of the application of arithmetic to geometry which teaches, from the actual measurement of certain lines of a figure, how to find, by calculation, the length of other lines, the area of surfaces, and the volume of solids. The determination of lines is, however, generally treated of under Trigonometry (q.v.), and surfaces and solids are now understood to form the sole subjects of mensuration. To find the length of a line (except in cases where the length may be calculated from other known lines, as in trigonometry) we have to apply the unit of length (in the shape of a footrule, a yard measure, a chain), and discover by actual trial how many units the line contains. But in measuring a surface or a solid we do not require to apply an actual square board, or a cubic block, or even to divide it into such squares or blocks; we have only to measure certain of its boundary lines or dimensions; and from them we can calculate or infer the contents. For example, to find the area of a rectangle it is sufficient to measure two adjacent sides and find the product of these in terms of the unit of length chosen; 7 feet 3 feet = 21 square feet.
The areas of other figures are found from this, by the aid of certain relations or properties of those figures demonstrated by pure geometry; for instance, the area of a parallelogram is the same as the area of a rectangle having the same base and altitude, and is therefore equal to the base multiplied by the height. As a triangle is half of a parallelogram, the rule for its area can be at once deduced. Irregular quadrilaterals and polygons are measured by dividing them into triangles, the area of each of which is separately calculated. For the area of the circle, see CIRCLE. The volume of a rectangular parallelepiped is found in cubic inches by multiplying together the length, breadth, and depth in inches; and the oblique parallelepiped, prism, or cylinder, by multiplying the area of the base by the height.