Convex Surface of a Frustum of a Pyramid or Cone.

are all equal; and the area of each is half the product of the sum of its parallel sides by their common altitude HH. The sum of their areas, or the area of the convex surface of the frustum is, therefore, half the product of this common altitude, by the sum of all the parallel sides, that is, by the sum of the perimeters of the bases of the frustum.

396. Corollary. If a section M'N'O'P' &c. is made by a plane parallel to the bases, and passing through the middle point R' of the altitude, it must, by § 336, bisect the lines AM, BN, &c.; and the area of each trapezoid is, by § 255, the product of its altitude by the line M'N', NO', &c.

The area of the convex surface of the frustum is, therefore, the product of the altitude by the sum of these lines, that is, by the perimeter of the section made by the plane which bisects the lateral sides of the frustum.

397. Corollary. The area of the convex surface of the frustum of a right cone is half the product of its side by the sum of the circumferences of the bases; or it is the product of the side by the circumference of the section parallel to the bases which bisects the side.

398. Theorem. The area of the surface, described by a line revolving about another line in the same plane with it as an axis, is the product of the revolving line by the circumference described by its middle point.

Proof. a. If the revolving line is parallel to the axis, as in (fig. 166), it describes the convex surface of a right cylinder, the area of which is, by § 353, the product of

Surface described by a revolving Line.

the circumference of the base by the altitude. But the altitude is equal to the revolving line, and the circumference of the base is, by § 354, equal to the circumference described by the middle point; and, therefore, in this case, the area of the surface described is the product of the revolving line by the circumference described by its middle point.

b. If the revolving line is inclined to the axis without meeting it, the surface described is the convex surface of the frustum of a right cone; and its area is as, in § 397, the product of the revolving line by the circumference described by its middle point.

c. When the revolving line meets the axis without cutting, the surface described is the convex surface of a right cone, and is included in the preceding case by considering it as a frustum whose upper base is the vertex of the

cone.

399. Scholium. The case, where the revolving line cuts the axis, is not included in the preceding theorem.

400. Theorem. The frustum of a pyramid or cone is equivalent to the sum of three pyramids or cones, which have for their common altitude the altitude of the frustum, and whose bases are the lower base of the frustum, its upper base, and a mean proportional between them.

Proof. Let ABCD &c. MNOP &c. (fig. 171) be the given frustum. Denote the area of the lower base ABCD &c. by V, and that of the upper base MNOP &c. by V; and denote the altitude ST of the greater pyramid by H, the altitude SR of the less pyramid by H', and the altitude RT of the frustum by H".

Since the frustum is the difference between the pyramids, we have for its solidity, by § 384,

1

Solids of a Frustum of a Pyramid or Cone.

} V × H— ¦ V' × H',

and, for the sum of three pyramids, which have H" for their altitude and for their bases V, V and the mean pro-` portional VV between V and V',

} H'' × (▼ + − + √VV)

H"

= } H′′ × V + } H′′ × V' + 3 H′′ × √VV, and we are to prove that these solidities are equal, or that V × H— V1× H=V× H' +V× H' + √VV × H". Now

and, by § 379,

H=H-H,

we obtain, by transposing the members of the first pro

duct,

√VV × H= V × H', ...

√VV' × H = V × H;

the difference between which is

VV × (H-H)=V× H-V' H, or

(H—H')

✔VV × H" = V × H' — V' × H.

And if we add to this last equation the equations

we get, by cancelling the terms which destroy each other,

VV XH +V× H" +V' × H-VX H-VXH, which is the equation to be proved, and the solidity of the frustum is therefore equal to

}· H" × (▼ + V' + √VV).

Solidity of the Frustum of a Cone.

401. Corollary. If R is the radius of the lower base of the frustum of a cone, and R' the radius of the upper base, we have, by § 280,

and the solidity of the frustum is

} π × H" × (R2 + R12 + R × R').

402. Scholium. The solidity of any polyedron may be found by dividing it into pyramids.

CHAPTER XVII.

403. Definition.

SIMILAR SOLIDS.

Similar polyedrons are those in which the homologous solid angles are equal, and the homologous faces are similar polygons.

404. Corollary. Hence, from § 170, the sides of similar polyedrons are proportional to each other.

405. Corollary. From § 268, the faces of similar polyedrons are to each other as the square of their homologous sides; and, from the theory of proportions, the sums of the faces, or the entire surfaces of the polyedrons are also to each other as the squares of the homologous sides.

Ratios of Similar Prisms, &c.

406. Corollary.

The bases of similar prisms or pyramids are to each other as the squares of their altitudes; and the perimeters of their bases are to each other as their altitudes.

407. Corollary. The bases of similar cylinders or cones are to each other as the squares of their altitudes; and their altitudes are to each other as the circumferences of the bases, or as the radii of the bases.

408. Corollary. The convex surfaces of similar prisms, pyramids, cylinders, or cones are to each other as their bases, or as the squares of their altitudes.

409. Corollary. The convex surfaces of similar prisms or pyramids are to each other as the squares of their homologous sides.

410. Corollary. The convex surfaces of similar cylinders or cones are to each other as the squares of the radii of their bases.

411. Theorem. Similar prisms, pyramids, cylinders, or cones are to each other as the cubes of their altitudes.

Proof. Prisms, pyramids, cylinders, or cones are to cach other, by § 366 and 386, as the products of their bases by their altitudes. But where these solids are similar, their bases are to each other, by § 406 and 407, as the squares of their altitudes; and the products of the bases by their altitudes, or their solidities are to each other, as the products of the squares of their altitudes by their altitudes, or as the cubes of their altitudes.

412. Corollary. Similar prisms or pyramids are to each other as the cubes of their homologous sides.