A # mensuration

Mensuration is the branch of geometry dealing with the measurement of point length, area, and volume. The base of all such measurements is length, since the areas and volumes of geometric figures can be calculated from suitable length measurements. The area of a rectangle is bh, where b is the length of one side (the base) and h that of the other (the height). It is easy to show that this formula also holds for the parallelogram, if h stands for the altitude (the perpendicular distance from one side to that facing it) rather than for the height; from this can be found the formula for the area of a triangle (which can be thought of as half a parallelogram), ½bh; and those for other polygons.

The area and circumference of a circle, which can be considered as a regular polygon with an infinite number of infinitely small sides, are πr2 and 2πr respectively, where π is a constant and r is the radius. The area of an ellipse is given by πab where a is the semimajor and b is the semiminor; the circumference of an ellipse cannot be expressed in algebraic terms.

From this information, it is fairly easy to determine formulae for the volumes of regular solids, such as the polyhedron, cone, cylinder, ellipsoid, pyramid, and sphere. The areas of irregular plane shapes can be approximated by considering a large number of extremely small strips, each being almost trapezoidal, formed in them by the construction of a large number of parallel chords; the same principle can be applied to finding the approximate volumes of irregular solids: this process is akin to integral calculus.