430. Similar triangular prisms are to each other as the cubes of their homologous edges.

Let CBD-P and C'B'D'-P' be two similar prisms, and let BC and B'C' be any two homologous edges.

To prove that

CBD-P: CB'D'-P' = BC: B'C3. .

Since the homologous angles B and B' are equal, and the faces which bound them are (by 404) similar, these triedral angles may be applied, one to the other, so that the angle C'B'D' will coincide with CBD, with the edge B'A' on BA.

In this case the prism C'B'D'-P' will take the position of cBd-p.

From A draw AH perpendicular to the common base of the prisms; then the plane BAH is (by 365) perpendicular to the plane of the base.

From a draw ah likewise in the plane BAH, and it will (by 364) be perpendicular to the plane of the base.

Since the bases BCD and Bcd are similar (by 262),

In the similar triangles ABH and aBh (by 218),

(a)

In the similar parallelograms AC and ac (by 224),

Multiplying (a) by (b), we have

CBD × AH: cBd × ah = CB: CB3.

But (by 428) CBD × AH is the volume of CBD-P, cBd x ah is the volume of C'B'D'-P', and cB = C'B'.

Therefore CBD-P : C'B'D'-P' – CB3 : C'B'3,

=

(b)

and

Q.E.D.

431. COR. 1. Any two similar prisms are to each other as the cubes of their homologous edges.

For, since the prisms are similar, their bases are similar polygons (by 404); and these similar polygons may each be divided into the same number of similar triangles, similarly placed (by 121); therefore, each prism may be divided into the same number of triangular prisms, having their faces similar and like placed; consequently, the triangular prisms are similar (by 404). But these triangular prisms are to each other as the cubes of their homologous edges, and being like parts of the polygonal prisms, the polygonal prisms themselves are to each other as the cubes of their homologous edges.

432. COR. 2. Similar prisms are to each other as the cubes of their altitudes, or as the cubes of any other homologous lines.

433. COR. 3. Since the cylinder is the limit of a prism of infinite number of sides, it follows that:

The volume of a cylinder is equal to the product of its base and altitude.

434. COR. 4. The volumes of two prisms (cylinders) are to each other as the product of their bases and altitudes: prisms

(cylinders) having equivalent bases are to each other as their altitudes: prisms (cylinders) having equal altitudes are to each other as their bases: prisms (cylinders) having equivalent bases and equal altitudes are equivalent.

EXERCISES.

Find the lateral area and volume:

1. Of a triangular prism, each side of whose base is 3, and whose altitude is 8.

2. Of a regular hexagonal prism, each side of whose base is 2, and whose altitude is 12.

3. Of a triangular prism whose altitude is 18 and the sides of the base are 6, 8, and 10.

PYRAMIDS.

435. A Pyramid is a polyedron, one of whose faces is a polygon, and whose other faces are triangles having a common vertex without the base and the sides of the polygon for bases. V

437. The Altitude of a pyramid is the perpendicular distance from the vertex to the plane of the base.

438. A pyramid is called Triangular, Quadrangular, Pentagonal, etc., according as its base is a triangle, quadrilateral, pentagon, etc.

439. A triangular pyramid has but four faces, and is called a Tetraedron; any one of its faces can be taken for its base.

440. A Regular Pyramid is one whose base is a regular polygon, the centre of which coincides with the foot of the perpendicular let fall upon it from the vertex. The lateral edges of a regular pyramid are (by 334) equal, hence the lateral faces are equal isosceles triangles.

441. The Slant Height of a regular pyramid is the altitude of any one of its lateral faces; that is, the straight line drawn from the vertex of the pyramid to the middle point of any side of the base.

442. A Truncated Pyramid is the portion of a pyramid included between its base and a plane cutting all the lateral edges.

443. A Frustum of a pyramid is a truncated pyramid whose bases are parallel.

The Altitude of a frustum is the perpendicular distance between the planes of its bases.

444. The lateral faces of a frustum of a regular pyramid are equal trapezoids.

The Slant Height of a frustum of a regular pyramid is the altitude of any one of its lateral faces.

445. A. Conical Surface is traced by a straight line so moving that it always intersects a given curve and passes through a given point.

Thus the straight line BB' continually intersects the curve ABC and passes through the point 0, tracing the conical A surface ABC–O–A'B'C'.

C

B'

B

446. The straight line BB' is the Generatrix, the curve ABC the Directrix, O the Vertex, and O-ABC, O–A'B'C' are the two Nappes, and OB is an Element.

447. A Cone is a solid bounded by a conical surface and a plane which cuts all of the elements of the surface, as O-ABC.

448. This plane is called the Base, and the perpendicular from the vertex to the plane of the base is the Altitude.

449. A Circular Cone is one whose base is a circle. NOTE. Hereafter Cone will be used for circular cone.

450. A Right Cone is a cone in which the perpendicular let fall from the vertex meets the base in its centre; it is also called a cone of revolution, since it can be formed by revolving a right triangle about one of its A shorter sides, as V-ABC.

C

B

451. Since the cone has a circular base which is the limit of a polygonal base, a cone may be regarded as a pyramid of an infinite number of faces, hence the cone will have, in general, the properties of a pyramid, and all demonstrations for pyramids will include cones when so stated in the theorem or in the corollary.

452. A Truncated Cone is the portion of a cone included between its base and another plane cutting all its elements.

453. A Frustum of a cone is a truncated cone whose cutting planes or bases are parallel.

The Altitude of a frustum is the perpendicular distance between the planes of its bases.