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The formula for the volume of a pyramidal square frustum was introduced by the ancient Egyptian mathematics in what is called the Moscow Mathematical Papyrus, written in the 13th dynasty (c. 1850 BC): = (+ +), where a and b are the base and top side lengths, and h is the height.
The formula for the volume of a frustum of a paraboloid [23] [24] is: V = (π h/2)(r 1 2 + r 2 2), where h = height of the frustum, r 1 is the radius of the base of the frustum, and r 2 is the radius of the top of the frustum. This allows us to use a paraboloid frustum where that form appears more appropriate than a cone.
The fourteenth problem of the Moscow Mathematical calculates the volume of a frustum. Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown. The volume is found to be 56 cubic units, which is correct. [1]
The volume of each segment is calculated as the volume of a frustum of a cone where: Volume= h(π/3)(r 1 2 + r 2 2 +r 1 r 2) Frustum of a cone. A similar, but more complex formula can be used where the trunk is significantly more elliptical in shape where the lengths of the major and minor axis of the ellipse are measured at the top and bottom ...
A spherical segment Pair of parallel planes intersecting a sphere forming a spherical segment (i.e., a spherical frustum) Terminology for spherical segments.. In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes.
The volume is equal to the product of the height of the frustum and the Heronian mean of the areas of the opposing parallel faces. [2] A version of this formula, for square frusta, appears in the Moscow Mathematical Papyrus from Ancient Egyptian mathematics, whose content dates to roughly 1850 BC. [1] [3]
For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 BC contained the first statements of the theorem; the Egyptians had a correct formula for the volume of a frustum of a square pyramid.
Rectangular (Cuboid): Several problems in the Moscow Mathematical Papyrus (problem 14) and in the Rhind Mathematical Papyrus (numbers 44, 45, 46) compute the volume of a rectangular granary. [13] Truncated pyramid (frustum) Frustum: The volume of a truncated pyramid is computed in MMP 14. [8]