prismoid's Usage Examples:

If R and S are the ends of a prismoid, A and B their areas, h the perpendicular distance between them, and C the area of a section by a plane parallel to R and S and midway between them, the volume of the prismoid is *h(A+4C+B).

37 The mensuration of earthwork involves consideration of quadrilaterals whose dimensions are given by special data, and of prismoids whose sections are D such quadrilaterals.

(iv) For the prismoid in general let ABCD .

cot Then volume of prismoid length X If mini + m2n2 + 2 (m i ne + m270-3a 2 1 tan 0.

The most instructive is to regard the prismoid as built up (by addition or subtraction) of simpler figures, which are particular cases of it.

In the ordinary case three of the four lateral surfaces of the prismoid are at right angles to the two ends.

Then the prismoid is divided into a pyramid with vertex P and base ABCD .

This formula applies to such figures as the cone, the sphere, the ellipsoid and the prismoid.

By (i) and (iii), the formula holds for each of these figures; and therefore it holds for the prismoid as a whole.

- 2 To show that the area of a cross-section of a - prismoid is of the form ax e -{- bx -{- c, where x is the distance of the section from one end, we may proceed as in § 27.

Another method of verifying the formula is to take a point Q in the mid-section, and divide up the prismoid into two pyramids with vertex Q and bases ABCD .