PROP. XIII. THEOREM.

499. The volume of any prism is equal to the product of its base and altitude.

Given any prism.

To Prove its volume equal to the product of its base and altitude.

Proof. The prism may be divided into triangular prisms by passing planes through one of the lateral edges and the corresponding diagonals of the base.

The volume of each triangular prism is equal to the product of its base and altitude.

(§ 498)

Then, the sum of the volumes of the triangular prisms is equal to the sum of their bases multiplied by their common altitude.

Therefore, the volume of the given prism is equal to the product of its base and altitude.

500. Cor. I. Two prisms having equivalent bases and equal altitudes are equivalent.

501. Cor. II. 1. Two prisms having equal altitudes are to each other as their bases.

2. Two prisms having equivalent bases are to each other as their altitudes.

3. Any two prisms are to each other as the products of their bases by their altitudes.

Ex. 14. Find the lateral area and volume of a regular hexagonal prism, each side of whose base is 3, and whose altitude is 9.

PYRAMIDS.

DEFINITIONS.

502. A pyramid is a polyedron bounded by a polygon, called the base, and a series of triangles

having a common vertex.

The common vertex of the triangular faces is called the vertex of the pyramid.

The triangular faces are called the lateral faces, and the edges terminating at the vertex the lateral edges.

The sum of the areas of the lateral faces is called the lateral area.

The altitude is the perpendicular distance from the vertex to the plane of the base.

503. A pyramid is called triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc.

504. A regular pyramid is a pyramid whose base is a regular polygon, and whose vertex lies in the perpendicular erected at the centre of the base.

505. A truncated pyramid is a portion of

a pyramid included between the base and a plane cutting all the lateral edges.

The base of the pyramid and the section made by the plane are called the bases of the truncated pyramid.

506. A frustum of a pyramid is a truncated pyramid whose bases are parallel. The altitude is the perpendicular distance between the planes of the bases.

EXERCISES.

4

15. Find the length of the diagonal of a rectangular parallelopiped whose dimensions are 8, 9, and 12.

16. The diagonal of a cube is equal to its edge multiplied by √3.

PROP. XIV. THEOREM.

507. In a regular pyramid,

I. The lateral edges are equal.

II. The lateral faces are equal isosceles triangles.

D

B

(The theorem follows by §§ 406, I, and 69.)

508. Def. The slant height of a regular pyramid is the altitude of any lateral face.

Or, it is the line drawn from the vertex of the pyramid to the middle point of any side of the base.

PROP. XV. THEOREM.

(§ 94, I)

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

D'

B' E

B

Given AC' a frustum of regular pyramid O-ABCDE.

To Prove faces AB' and BC' equal trapezoids.
Proof. We have AOAB = = A OBC.

(§ 507, II)

We may then apply ▲ OAB to ▲ OBC in such a way that sides OB, OA, and AB shall coincide with sides OB, OC, and BC, respectively.

Now, A'B' || AB and B'C' || BC.

Hence, line A'B' will coincide with line B'C'.

(?)

(§ 53)

Then, AB' and BC' coincide throughout, and are equal.

510. Cor. The lateral edges of a frustum of a regular pyramid are equal.

511. Def. The slant height of a frustum of a regular pyramid is the altitude of any lateral face.

PROP. XVI. THEOREM.

512. The lateral area of a regular pyramid is equal to the perimeter of its base multiplied by one-half its slant height.

H
B

Given slant height OH of regular pyramid O-ABCDE.

To Prove

lat. area 0-ABCDE =

(AB+ BC + etc.) × 1 OH.

(By § 508, OH is the altitude of each lateral face.)

513. Cor. The lateral area of a frustum of a regular

pyramid is equal to 'one-half the sum of

E'

the perimeters of its bases, multiplied by

D'

its slant height.

H

B

Given slant height HH' of the frus- 4 tum of a regular pyramid AD'.

E

H

B

} (AB + A'B' + BC + B'C' + etc.) × HH'.

(HH' is the altitude of each lateral face.)

EXERCISES.

17. The volume of a cube is 417 cu. ft. Find the area of its entire surface in square inches.

18. The volume of a right prism is 2310, and its base is a right triangle whose legs are 20 and 21, respectively. Find its lateral area. 19. Find the lateral area and volume of a right triangular prism, having the sides of its base 4, 7, and 9, respectively, and the altitude 8. 20. The volume of a regular triangular prism is 96 √3, and one side of its base is 8. Find its lateral area.

21. The diagonal of a cube is 8 √3. Find its volume, and the area of its entire surface.

(Represent the edge by x.)

22. A trench is 124 ft. long, 2 ft. deep, 6 ft. wide at the top, and 5 ft. wide at the bottom. How many cubic feet of water will it contain? (§§ 316, 499.)

23. The lateral area and volume of a regular hexagonal prism are 60 and 15 √3, respectively. Find its altitude, and one side of its base. (Represent the altitude by x, and the side of the base by y.)

PROP. XVII. THEOREM.

514. If a pyramid be cut by a plane parallel to its base, I. The lateral edges and the altitude are divided proportionally.

II. The section is similar to the base.

Given plane A'C' to base of pyramid O-ABCD, cutting faces OAB, OBC, OCD, and ODA in lines A'B', B'C', C'D', and D'A', respectively, and altitude OP at P'.