Pyramid (geometry) - Volume

Volume

See also: Cone (geometry) – Volume

The volume of a pyramid (also any cone) is where b is the area of the base and h the height from the base to the apex. This works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. In 499 AD Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.6).

The formula can be formally proved using calculus: By similarity, the linear dimensions of a cross section parallel to the base increase linearly from the apex to the base. The scaling factor (proportionality factor) is, or, where h is the height and y is the perpendicular distance from the plane of the base to the cross-section. Since the area of any cross section is proportional to the square of the shape's scaling factor, the area of a cross section at height y is B×, or since both b and h are constants . The volume is given by the integral

The same equation, also holds for cones with any base. This can be proven by an argument similar to the one above; see volume of a cone.

For example, the volume of a pyramid whose base is an n-sided regular polygon with side length s and whose height is h is:

Read more about this topic:  Pyramid (geometry)

Famous quotes containing the word volume:

    “A big leather-bound volume makes an ideal razorstrap. A thin book is useful to stick under a table with a broken caster to steady it. A large, flat atlas can be used to cover a window with a broken pane. And a thick, old-fashioned heavy book with a clasp is the finest thing in the world to throw at a noisy cat.”
    —Mark Twain [Samuel Langhorne Clemens] (1835–1910)

    “Thy commandment all alone shall live
    Within the book and volume of my brain
    Unmixed with baser matter.”
    —William Shakespeare (1564–1616)

    “Although volume upon volume is written to prove slavery a very good thing, we never hear of the man who wishes to take the good of it, by being a slave himself.”
    —Abraham Lincoln (1809–1865)