Exercises. Properties of Pyramids

1. The volume of a triangular pyramid is equal to one third the volume of a triangular prism of the same base and altitude.

2. The volumes of two pyramids are to each other as the products of their bases and altitudes.

3. Pyramids with equivalent bases are to each other as their altitudes.

4. Pyramids with equal altitudes are to each other as their bases.

5. Pyramids with equivalent bases and equal altitudes are equivalent.

6. In the tetrahedron (§ 114) V-ABC, the midpoints of VA, VC, BA, BC are vertices of a parallelogram.

7. The lines joining the midpoints of the opposite edges of a tetrahedron meet in a point.

In the figure described in Ex. 6 the opposite edges are VA and BC; VB and AC; VC and AB.

8. The plane which passes through an edge of a tetrahedron and the midpoint of the opposite edge divides the tetrahedron into two equivalent tetrahedrons.

9. Given that a regular triangular pyramid has all its four faces congruent, and that its volume is known, show how to find the area of the total surface.

10. Show how to find the volume of any polyhedron by dividing the polyhedron into pyramids.

11. Given the edge a of the base and the area T of the total surface of a regular pyramid with a square base, find the height h in terms of a and T.

12. In Ex. 11 find the volume V in terms of a and T.

Exercises. Measuring the Pyramid

1. What is the lateral area of a regular pyramid whose slant height is 34 in., and the perimeter of the base 57 in.?

2. Find the volume of a pyramid with an altitude of 7 in. and a base 9 sq. in. in area.

3. The base of a regular pyramid is an octagon 3 m. on a side and the slant height is 5 m. Find the lateral area of the pyramid.

4. Find the volume of a pyramid with an altitude of 6.75 m. and a square base whose diagonal is 3 √2 m.

5. The volume of a regular pyramid with a square base is 912 cu. ft. and the altitude is 19 ft. Find the lateral area. 6. The volume of a regular pyramid with a hexagonal base is 249.4 cu. m., and the altitude is 8 m. Find the length of each side of the base.

7. The base of a pyramid is a triangle with sides which are 6 in., 8 in., and 10 in., and the volume is 240 cu. in. Find the height of the pyramid.

8. A pyramid 12 in. high has a base which is an equilateral triangle 10 in. on a side. Find the volume.

9. Find the volume of a regular pyramid with a lateral edge of 100 ft. and a square base whose side is 40 ft.

10. Find the volume of a regular pyramid whose slant height is 12 ft. and whose base is an equilateral triangle inscribed in a circle 10 ft. in radius.

11. The eight edges of a regular pyramid with a square base are equal and the area of the total surface is T. Find the edge.

12. Given the height h and the area Tof the total surface, find the base edge a of a regular quadrangular pyramid.

Proposition 15. Volume of a Frustum

125. Theorem. The volume of a frustum of a pyramid is one third the product of the altitude by the sum of the two bases and the mean proportional between them.

F

h

Given F, a frustum; V, the volume; B, B', the areas of the bases; and h, the altitude.

Prove that V = {} h(B+B' +√ BB').

Proof. Let P be the volume of the pyramid from which F is cut, and let P' and h' be the volume and altitude respectively of the small pyramid remaining after F is removed. V = P — P' = {} B (h + h') — 3 B'h'.

$ 124

√B:√B'=h+h': h'. § 120, 3; Ax. 6 h√B'(√B+√B')

Then

Now

Solving, h'=

h√B'

Then

V = } | Bh + (B — B') h √ B'(√B+√B'

Ax. 5

B-B'

Simplifying, V=}h(B+B'+√BB').

This is a subsidiary proposition in mensuration and may be omitted, together with pages 89 and 90, without disturbing the sequence.

Exercises. Frustum of a Pyramid

1. A piece of marble is in the form of a frustum of a regular pyramid with a square base. The frustum is 8 ft. high and the sides of the bases are 3 ft. and 2 ft. respectively. Taking the weight of 1 cu. ft. of marble as 165 lb., find the weight of the piece.

2. The slant height of a frustum of a regular pyramid is 10 ft. and the sides of the square bases are 3 ft. and 2 ft. respectively. Find the area of the total surface.

3. How much earth was removed in an excavation which is 6 ft. deep, 40 ft. square at the top, and 36 ft. square at the bottom?

4. A pile of broken stone is in the form of a frustum of a pyramid. The lower base is a rectangle 75 ft. long and 9 ft. wide, the upper base is 50 ft. by 6 ft., and the height of the frustum is 6 ft. If the broken stone is spread over a road 30 ft. wide to a depth of 3 in., what length of road will it cover?

5. A pyramid 4 in. high with a base whose area is 16 sq. in. is cut by a plane parallel to the base and 2 in. from the vertex. Find the volume of the frustum.

6. A pyramid 6 in. high with a base whose area is 324 sq. in. is cut by a plane parallel to the base and 2 in. from the vertex. Find the volume of the frustum.

7. The lower base of a frustum of a pyramid is a square 8 in. on a side. The side of the upper base is half that of the lower base, and the altitude of the frustum is the same as the side of the upper base. Find the volume of the frustum.

8. Consider the formula Vh (B+B'+√BB') of § 125 when B'= 0. Discuss the meaning of the result. Also discuss the case in which B=B'.

Exercises. Review

1. The lower base of a frustum of a pyramid is a square 6 in. on a side. The area of the upper base is half that of the lower base, and the altitude of the frustum is 4 in. Find to the nearest 0.01 cu. in. the volume of the frustum.

2. A pyramid has six edges, each 2 in. long. Find to the nearest 0.01 cu. in. the volume of the pyramid.

3. A regular pyramid 8 in. high has a triangular base, and the volume of the pyramid is 16 √2 cu. in. Find to the nearest 0.01 in. the length of a side of the base.

4. In Ex. 3 find the area of the total surface.

5. The base of a regular pyramid is a square l feet on a side, and the slant height is s feet. Find the area of the total surface.

6. The lower base of a frustum of a pyramid is a quadrilateral whose sides are 5 in., 6 in., 7 in., and 9 in., respectively, and the corresponding sides of the upper base are 3 in., x inches, y inches, and z inches. Find x, y, and z.

7. A schoolroom 30 ft. long, 24 ft. wide, and 14 ft. high is ventilated by an electric fan which discharges every 20 min. a volume of air equal to the volume of the room. Find the amount of air discharged per minute.

8. An oblique pyramid with a square base 10 in. on a side is cut by a plane parallel to the base so that the altitude, measured from the vertex, is divided in the ratio 3:2. Find the area of the section.

9. A frustum of a square pyramid is 18 ft. high and the sides of the bases are 1 ft. and 3 ft. respectively. If the frustum is divided into three parts by planes passed parallel to the bases and dividing the altitude equally, what is the volume of each part?