If a pyramid is cut through by a plane parallel to its base portion of the pyramid between that plane and the base is called frustum of the pyramid.
Volume of Frustum of a Pyramid: Consider any frustum of a pyramid in figure with the lower base, upper base and the altitude. Complete the pyramid of which the frustum is a part. Denote by, the volume of the small pyramid, whose altitude is. Then the altitude of is. Let and respectively, represents the volume of the frustum and the pyramid.
From the figure it is easily seen that. Expressing this equality in terms of the dimensions, we may write.
The pyramid may be considered as cut by the two parallel planes and. Hence
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Taking the square root of both sides, we have Transposing to the L.H.S. of this equation and factorizing,
Substituting the value of in (1), we have
i.e. The volume of a frustum of a pyramid is equal to the one-third the product of the altitude and the sum of the upper base, the lower base and the square root of the product of two bases.
Lateral Surface Area of Frustum of a Pyramid: In the frustum of pyramid on a square base, let denote the length of each side of the base, the length of each side of the other end, the height of the frustum.
Each face CDML is a trapezium, the lengths of the parallel sides and.
Example:
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