Section of Pyramid parallel to the Base.

altitude SH; that is, the convex surface of the pyramid is half the product of the perimeter of its base by the altitude of one of its triangles.

376. Corollary. When the base of the regular pyramid is a polygon of an infinite number of sides, the pyramid is a right cone, and the altitude of each triangle becomes the side SA (fig. 174) of the cone.

Hence the area of the convex surface of the right cone is half the product of the circumference of the base by the side.

377. Theorem. The section of a pyramid made by a plane parallel to the base is a polygon similar to the base.

Proof. Let MNOP &c. (fig. 171) be the section of a pyramid made by a plane parallel to its base ABCD &c. Since MN is, by § 334, parallel to AB, we have SB: SNAB : MN,

and since NO is parallel to BC, we have

SB: SN-BC : NO ;

and, on account of the common ratio, SB: SN, AB: MNBC: NO.

In the same way we might prove

AB: MN=BC : NO = CD: OP, &c.

whence the sides of the polygons ABCD &c., MNOP &c. are proportional.

The angles of the polygons are also equal; indeed on account of the parallel sides, we have

MNO ABC, NOP-BCD, &c.

=

The polygons are therefore similar, by § 170.

378. Corollary. The section of a cone made by a plane parallel to the base is a circle.

Equivalent Pyramids and Cones.

379. Corollary. If the perpendicular ST is let fall from S upon the base, meeting the section at R, we have, by SS 268 and 336,

or, the base of a pyramid or cone is, to the section made by a plane parallel to the base, as the square of the altitude of the pyramid is to the square of the distance of the section from the vertex.

380. Corollary. If two pyramids or cones have the same altitude and their bases in the same plane, their sections made by a plane parallel to the plane of their bases are to each other as their bases.

If the bases are equivalent, the sections are equivalent.

If the bases are equal, the sections are equal.

381. Theorem. Two pyramids or two cones which have equal bases and altitudes are equivalent.

Proof. For, if their bases are placed in the same plane, their sections made by a plane parallel to the plane of their bases are equal; and, therefore, by § 342, the pyramids are equivalent.

382. Theorem. A triangular pyramid is a third part of a triangular prism of the same base and altitude.

Proof. From the vertices B, C (fig. 175) of the triangular pyramid S ABC, draw BD, CE parallel to SA. Draw SD, SE parallel to AB, AC, and join CE; ABC SDE is a triangular prism.

The quadrangular pyramid S BCED is divided by the plane SBE into two triangular pyramids S BED, S BEC, which are equivalent; for their bases BED,

Solidity of the Pyramid and Cone.

BEC are equal, by § 77; and their common altitude is the distance of their common vertex S from the plane of their bases.

Again, if the plane SED is taken for the base of S BED and the point B for its vertex, the pyramid B SDE is equivalent to SABC; for their bases SED, ABC are equal, and their common altitude is the altitude of the prism.

But the sum of the three equal pyramids S ABC, S BED, S BEC is the prism ABC SDE, and, therefore, either pyramid, as S ABC, is a third part of the prism.

383. Corollary. The solidity of a triangular pyramid is a third of the product of its base by its altitude.

384. Theorem. The solidity of any pyramid is one third of the product of its base by its altitude.

Proof. The planes SAC, SAD, &c. (fig. 171) divide the pyramid S ABCD &c. into triangular pyramids, the common altitude of which is the altitude of the entire pyramid. Hence the solidity of the entire pyramid is one third of the product of the sum of their bases ABC, ACD, &c., by the common altitude, that is, one third of the entire base by the altitude of the pyramid.

385. Corollary. The solidity of a cone is one third of the product of its base by its altitude.

386. Corollary. Pyramids or cones are to each other as the products of their bases by their altitudes.

387. Corollary. Pyramids or cones of the same altitude are to each other as their bases; and those of equivalent bases are to each other as their altitudes.

388. Corollary. Pyramids or cones of equivalent bases and equal altitudes are equivalent.

Truncated Prism.

389. Corollary. Any pyramid or cone is a third part of a prism or cylinder of the same base and altitude.

390. Corollary. Denoting by R the radius of the base of a cone, by H its altitude, by Vits solidity, and using л, as in § 237, we have, by §§ 369 and 389,

391. Definitions. A truncated prism is the portion of a prism cut off by a plane inclined to its base, as ABC DEF (fig. 176).

The base of the truncated prism is the same as the base of the prism from which it is cut.

392. Theorem. A truncated triangular prism is equivalent to the sum of three pyramids, which have for their common base the base of the prism, and for their vertices the three vertices of the inclined section.

Proof. Draw the plane FAC (fig. 176), cutting off from the truncated triangular prism ABC DEF the pyramid FABC, which has ABC for its base, and F for its vertex.

There remains the quadrangular pyramid FACDE, which the plane FEC divides into the two triangular pyramids FAEC and FCDE.

Now FAEC is equivalent to the pyramid BAEC, which has the same base AEC, and the same altitude, because the vertices F, B are in the line FB parallel to this base. But ABC may be taken for the base of EABC, and E for its vertex.

Lastly,

the pyramid FECD-the pyramid BECD,

for they have the same base ECD, and the same altitude,

Frustum of a Pyramid or Cone.

because their vertices F, B are in the line FB parallel to this base. Also, taking E as the vertex of BECD

the pyramid EBCD=the pyramid ABCD,

for, they have the common base BDC, and their vertices A, E are in the line AE parallel to this base. But ABC may be taken for the base of ABCD, and D for its ver

tex.

and

Hence the truncated prism is equivalent to the sum of three pyramids, which have the common base ABC, for their vertices E, F, and D.

393. Definitions. If a pyramid or cone is cut by a plane parallel to its base, the portion which remains after taking away the smaller pyramid or cone, is called the frustum of a pyramid or cone, as ABCD &c. MNOP &c. (fig. 171.)

The convex surface of the frustum of a pyramid is the sum of the trapezoids which compose its lateral faces.

The polygons ABCD &c., MNOP &c. are the bases of the frustum, and the distance between its bases is its altitude.

394. Corollary. The frustum of the right cone (fig. 174) may be considered as generated by the revolution of the trapezoid 00'A'A about the side 00.

The side AA', which is called the side of the frustum, in this case, generates the convex surface.

395. Theorem. The area of the convex surface of the frustum of a regular pyramid is half the product of the sum of the perimeters of its bases by the altitude of either of its trapezoids.

Proof. The trapezoids ABMN, BCNO, &c. (fig. 172)