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).

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

(iv) For the **prismoid** in general let ABCD ...

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

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 ...

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

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

- 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.

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