PYRAMID 



Kfi 



PYRAMID 



altitude of each triangular face.) The area of 

 the entire lateral surface of a pyramid is th> 

 product of the perimeter of the base and one- 

 half the slant height. 



Volume, (a) Build of cardboard a prism and 

 a pyramid of the same dimensions. Fill the 

 pyramid with sand or sugar. Empty its con- 

 tents into the prism. Do this until the prism 

 is full. You will find that the prism holds 

 three times as 

 much as the pyra- 

 mid. (6) Cut a 

 prism and a pyra- 

 mid of the same 

 base and altitude, 

 from clay, putty 

 or from a potato. 

 Weigh both. You 

 will find that the 

 prism weighs 

 three times as 

 much as the pyra- 

 mid. 



The volume of 

 a pyramid is one- 

 third as great as 

 <>lume of a 

 prism of the 

 same dimen- 

 sions, or The volume of a pyramid is tin 

 n r i a of the base multiplied by one-third the 

 altitude. The formula is: Volume of pyramid 



v . altitude 



=area of baseX r 



o 



Problems. 1. How many square feet of sheet- 

 ing on the sides of a steeple, a square pyramid 

 in shape, with base 12 feel square and slant 

 height 20 feet? 



_ . , slant height 

 Lateral surface = perimeter x <j 



rnber of sq. ft.= (4Xl2) X 2 %=480 



2. What is the volume of a pyramid whose 

 base is 10 feet square and altitude 15 feet? 



Volume of pyramids area of basex alt> 8 tu< 

 :nc In cu. ft.= 



FIG. 1 



Explanation appears in the 

 text. 



:M! the lateral surface of a six-aided 

 pyramid whose slant height is 30 feet and each 

 "f whose base is 8 feet. 



4. What is the volume of a pyramid whose 

 base is a rectangle 10x15 feet, and whose alti- 

 tude is 21 feet? 



5. At $.45 a square foot, how much will it 

 cost to gild a five-sided pyramidal steeple if 



.- i- in ;'. . t on each side and the slant 

 n 16 feet? 



6. A pyramid has as its base a right triangle 

 with sides respectively 10 inches, 15 inches and 

 IS inches; its altitude is 24 inches. Find the 

 volume of the pyramid and the number of 

 square inches of lateral surface. 



Frustum of a Pyramid. The part of a pyra- 

 mid between the base and a plane which cuts 

 the pyramid parallel to the base is called a 

 frustum. (See Fig. 2.) 



Lateral Area. Its lateral area is made up of 

 trapezoids whose lower edges make the perime- 

 ter of the lower base of the frustum, and whose 

 upper edges make the perimeter of the upper 

 base of the frustum. To find the area of the 

 lateral surface of a frustum, multiply one-half 

 the sum of the perimeters of the bases by the 

 slant height. See TRAPEZIUM. 



Volume. To find the volume of a frustum, 

 multiply one-half the sum of the areas of the 

 two bases by the altitude (the distance be- 

 tween the centers of the two bases, or a c in 

 Fig. 2). 



Problems. 1. How many square inches of 

 sheet iron are 

 used in making a 

 bread pan 8 

 inches by 5 inches 

 at the bottom 

 and 9 inches by 6 

 inches at the top, 

 the slant height 

 being 2% inches? 

 (The reader will 

 note that it is 

 the frustum of a 

 rectangular pyra- 

 mid.) 



FIG. 2 



See Pig:. 1 for explanation 

 of symbols. 



Lateral surface In sq. In. = %X 214=70 

 Bottom surface In sq. In. = 8x6 = 40 

 Number sq. In. of sheet lron=110 



2. How many cubic inches in the volume of 

 a tin box whose lower base is 10 inches square. 

 whose upper base is 6 inches square and whose 

 altitude is 8 inches? 



Area of 2 baaes= (10x 10") + (6x6) =136 

 Volume In cu. In. = 



3. Find the area of the lateral surface of a 

 frustum of a regular pyramid whose lower 

 base is a square 10 feet on a side, upper base 

 5 feet on a side, and slant height 16 feet. 



4. A monument in the form of a frustum 

 of a square pyramid contains 445 cubic fc< 

 stone. The lower base is 8 feet on a side, the 

 upper base 5 feet on a side. What is tin 

 height of the monument? A.H. 



