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Exercises. Practical Measurements

1. If the length of a rectangular parallelepiped is 22 in., the width 8 in., and the height 6 in., what is the area of the total surface?

2. Find the volume of a triangular prism whose height is 14 in. and whose base has the sides 10 in., 8 in., and 6 in. In such cases the student may use any formula given in §§ 275–280.

3. Find the volume of a prism whose height is 12 ft. and whose base is an equilateral triangle 10 in. on a side.

4. The base of a right prism is a rhombus of which one side is 10 in., and the shorter diagonal 12 in. The height of the prism is 15 in. Find the area of the total surface and the volume of the prism.

5. An open tank 8 ft. long and 5 ft. wide holds 264 cu. ft. of water. How many square feet of sheet lead will it take to line the sides and bottom?

6. How much sheet lead is required to line an open tank which is 5 ft. long, 3 ft. 6 in. wide, and contains 105 cu. ft.?

7. The diagonal of one of the faces of a cube is √7 in. Find the volume of the cube.

8. The three dimensions of a rectangular parallelepiped are a, b, c. Find in terms of a, b, and c the volume, the area of the total surface, and the length of the diagonal.

9. If the height of a prism is 5 in., and the base is a regular hexagon 1 in. on a side, what is the volume?

10. An open cistern is made of iron in. thick, and the inside dimensions are as follows: length, 6 ft.; width, 4 ft.; depth, 3 ft. What will the cistern weigh when empty? when full of water?

A cubic foot of water weighs 62 lb. The specific gravity of iron is 7.2; that is, iron is 7.2 times as heavy as water.

II. PYRAMIDS

113. Pyramid. A polyhedron of which one face, called the base, is any polygon and the other faces are triangles with a common vertex is called a pyramid.

The triangular faces with the common vertex are called the lateral faces, their intersections are called the lateral edges, and their common vertex is called the vertex of the pyramid. In our work we shall consider only pyramids whose bases are convex polygons.

The sum of the areas of the lateral faces is called the lateral area of the pyramid. The perpendicular distance from the vertex to the plane of the base is called the height or altitude of the pyramid.

[graphic]

114. Pyramids classified as to Bases. Pyramids are said to be triangular, quadrangular, and so on, according as their bases are triangles, quadrilaterals, and so on.

A triangular pyramid is also called a tetrahedron.

115. Regular Pyramid. If the base of a pyramid is a regular polygon whose center coincides with the foot of the perpendicular from the vertex to the base, the pyramid is called a regular pyramid.

The altitude of the triangle which forms one of the lateral faces of a regular pyramid is called the slant height of the pyramid.

The figures above show different types of pyramids, the two at the right being regular pyramids.

116. Properties of Regular Pyramids. The proofs of the following obvious properties of regular pyramids depend upon the theorems indicated:

1. The lateral edges of a regular pyramid are

equal (§ 43).

For they meet the base at equal distances from the center (§ 115).

2. The lateral faces of a regular pyramid are congruent isosceles triangles (§ 7, 5).

3. The slant height of a regular pyramid is the same for all the lateral faces ($ 43).

117. Frustum of a Pyramid. The portion of a pyramid included between the base and a section parallel to the base is called a frustum of a pyramid.

The figure at the right shows a frustum of a regular pyramid.

A more general term, including a frustum as a special case, is truncated pyramid, which is applied to the portion of a pyramid inIcluded between the base and any section whatever made by a plane that cuts all the lateral edges. This term, however, is little used at the present time.

The base of the pyramid and the parallel section are called the bases of the frustum.

The perpendicular distance between the bases is called the height or altitude of the frustum. The altitude is represented by h in the figure here shown.

The portions of the lateral faces of a pyramid that lie between the bases of a frustum are called the lateral faces of the frustum, and the sum of their areas is called the lateral area of the frustum.

[graphic]

h

The altitude of one of the lateral faces of a frustum of a regular pyramid is called the slant height of the frustum. The slant height is represented by l in the above figure.

Proposition 10. Lateral Area of a Pyramid

118. Theorem. The lateral area of a regular pyramid is half the product of the slant height and the perimeter of the base.

B

C

Given V-ABCDE, a regular pyramid; L, the lateral area; 1, the slant height; and p, the perimeter of the base.

Prove that

L= 1⁄2lp.

Proof.

The lateral faces are congruent A.

§ 116, 2

and

Then

Now the area of each face is times the base, the sum of the bases of the A is p. the sum of the areas of the A is lp.

§ 25, 3

Ax. 1

[blocks in formation]

119. Corollary. The lateral area of a frustum of a regular

pyramid is half the product of the slant height of the frustum and the sum of the perimeters of the bases.

How is the area of a trapezoid found (§ 25, 4)? Are the faces congruent trapezoids? What is the sum of their lower bases? of their upper bases? What is the sum of their areas? Write the formula and give the proof in full,

D'

C'

B'

Α'

B

Proposition 11. Section Parallel to Base

120. Theorem. If a pyramid is cut by a plane parallel to the base,

1. The lateral edges and the altitude are divided proportionally.

2. The section is a polygon similar to the base.

3. The area of the section is to the area of the base as the square of the distance of the plane from the vertex is to the square of the altitude of the pyramid.

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Given the pyramid V-ABCDE cut by m, a plane to the base and intersecting the altitude VO in O', and A'B'C'D'E', the section thus formed.

[blocks in formation]

1. Use §§ 55 and 22, 5 to prove the first proportion.

2. Prove AVA'B' similar to AVA B, and so on. Then prove the neces sary conditions (§ 24, 1) under which two polygons are similar.

3. Prove that A'B'C'D'E': ABCDE=A'B': AB2=VO2: VO2.

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