Pyramid.

FGH, ACD FHI &c. by the planes ACFH, ADFI &c., and, by the preceding article, the solidity of each of these triangular prisms is the product of its base ABC, ACD &c. by the altitude PQ. Hence, the sum of these prisms or the entire prism is the product of the sum of the bases by PQ, or of the entire base ABCD &c. by the altitude PQ.

b. This demonstration is extended to cylinders by increasing the number of sides to infinity.

367. Corollary. Prisms or cylinders of equivalent bases and equal altitudes are equivalent.

368. Corollary. Prisms or cylinders of equivalent bases are to each other as their altitudes; and those of the same altitude are to each other as their bases.

369. Corollary. Denoting by R the radius, and by A the area of the base of a cylinder; and using as in art. 237, we have, by art. 280,

Denoting, also, by H the altitude, V the solidity of the cylinder, we have, by art. 366,

V=AX H = π × R2 × H.

370. Definitions. A pyramid is a solid formed by several triangular planes proceeding from the same point, and terminating in the sides of a polygon, as SABCD &c. (fig. 171)

The point S is the vertex of the pyramid.

The polygon ABCD &c. is the base of the pyramid.

The convex surface of the pyramid is the sum of the triangles SAB + SAC &c.

Cone, Convex Surface of the Regular Pyramid, Right Cone.

The altitude of the pyramid is the distance of its vertex from its base.

371. Definitions. A pyramid is triangular, quadrangular, &c., according as the base is a triangle, a quadrilateral, &c.

372. Definitions. A pyramid is regular, when the base is a regular polygon, and the perpendicular let fall from the vertex upon the base, passes through the centre of the base (fig. 172).

This perpendicular from the vertex is called the axis of the pyramid.

373. Definitions. When the base of a pyramid is a regular polygon of an infinite number of sides, that is, a circle, it is called a cone (fig. 173).

The axis of the cone is the line drawn from the vertex to the centre of the base.

A right cone is one the axis of which is perpendicular to the base (fig. 174).

374. Corollary. The right cone (fig. 174) may be considered as generated by the revolution of the right triangle SOA about the axis SO.

The leg OA, in this case, generates the base, and the hypothenuse SA, which is called the side of the cone, generates the convex surface.

375. Theorem. The area of the convex surface of the regular pyramid is half the product of the perimeter of the base by the altitude of one of the triangles.

Section of a Pyramid.

Demonstration. The triangles SAB, SBC &c. (fig. 172) are all equal, for, by art. 201,

and, since the oblique lines SA, SB, SC, &c., are all a equal distances OA, OB, OC &c., from the perpendicular SO, they are equal by art. 319. Hence the altitudes SH, SI, SK, &c. of these triangles are equal; and the sum of the areas of the triangles is half the product of the sum of their bases AB, BC, CD, &c. by the common altitude SM; 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 the product of the circumference of the base. X 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.

Demonstration. 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 art. 333, parallel to AB, we have

SB: SNAB : MN,

and since NO is parallel to BC, we have

SB: SN = BC : NO;

Section of a Pyramid and Cone.

and, on account of the common ratio, SB: SN,

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 art. 169.

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

379. Corollary. If the perpendicular ST is let fall from S upon the base, meeting the section at R, we have, by arts. 268 and 335,

ABCD &c. : MNOP &c. = AB2: MN2 = SA2: SM2

= ST2: SR2,

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.

Solidity of Pyramid and Cone.

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

Demonstration. 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 art. 342, the pyramids are equivalent.

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

Demonstration. 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 BE; 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, BEC are equal, by art. 76; 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 S.. ABC; 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.