ured, and if the quotient is taken in each case, the d quotients will be found to have nearly the same value. If absolutely correct measurements could be made, the quotient in each case would be the same and equal to 3.14159+, i.e. the ratio of the circumference of a circle to its diameter is the same for all circles. The ratio is denoted by the Greek letter π (pi). The value of π found in geometry is 3.14159+. In common practice π is taken as 3.1416. The value of T cannot be exactly expressed by any number, but can be found correct to any desired number of decimal places. с 193. Since =π and d d = 2r, where r stands for the C radius of the circle, then cπd=2πr, and d= and r = с 27 π 2 Hence, if the radius, diameter or circumference of a circle is known, the other parts can be found. Thus, the circumference of a circle whose radius is 10 in. is 2 x 3.1416 x 10 in. = = 62.832 in. EXERCISE 30 1. Find the circumference of a circle whose diameter is 20 in. 2. Find the radius of a circle whose circumference is 250 ft. 3. If the length of a degree of the earth's meridian is 69.1 mi., what is the diameter of the earth? 4. If the radius of a circle is 8 in., what is the length of an arc of 15° 20' ? 5. The diameter of a circle is 10 ft. How many degrees are there in an arc of 16 ft. long? 194. The Area of a Circle. The circle may be divided into a number of equal figures that are essentially triangles. The sum of the bases of . these triangles is the circumference of the circle, and the altitudes are radii of the circle. Treating these figures as triangles, their areas will be cxr. Therefore, since c=2πr, A=1 of 2 πr xr=r2. It is proved in geometry that this result is exactly correct. The area of a circle whose radius is 5 ft. is 3.1416 × 5 × 5 sq. ft. 78.54 sq. ft. EXERCISE 31 1. Find the area of a circle whose radius is 10 in. = 2. Find the area of a circle whose circumference is 25 ft. 3. Find the radius of a circle whose area is 100 sq. ft. 4. The areas of two circles are 60 sq. ft. and 100 sq. ft. Find the number of degrees in an arc of the first that is equal in length to an arc of 45° in the second. 5. Find the side of a square that is equal to a circle whose circumference is 50 in. longer than its diameter. B 195. The Volume of a Rectangular Parallelopiped. If the unit of measure a is 1 cu. in., then the column AB is 4 cu. in., and the whole section ABCD will · contain 3 of these columns, or 3 × 4 cu. in. Since there are five of these sections in the parallelopiped, the entire volume (V) is 5 × 3 × 4 cu. in., or 60 cu. in. Therefore, the vol 5 3 a A ume of a rectangular parallelopiped is equal to the products of its three dimensions. That is, the number of cubic units in the volume is equal to the product of the three numbers that represent its dimensions. 196. If the dimensions of the rectangular parallelopiped are a, b and c, it can be shown in the same way that V=abc. Any of these four quantities, V, a, b, c, can be determined when the other three are known. Ex. If the volume of a rectangular parallelopiped is 36 cu. in. and two of the dimensions are 6 in. and 2 in., the third dimension is 36 3. .. 3 in. is the other dimension. 6 × 2 = 197. If the dimensions of a rectangular parallelopiped are equal, the figure is a cube and the volume is equal to the third power of the number denoting the length of its edge (a), or V= a3. For this reason the third power of a number is called its cube. 198. It is proved in geometry that any parallelopiped has the same volume as a rectangular parallelopiped with the same base and altitude. EXERCISE 32 1. Find the volume of a cube 3 in. on an edge. 2. Find the volume of a rectangular parallelopiped whose edges are 3cm, 5cm and 11cm. 3. The volume of a rectangular parallelopiped is 100 cu. in. The area of one end is 20 sq. in. Find the length. 4. How many cubic feet of air are there in a room 12 ft. 6 in. long, 10 ft. 8 in. wide and 9 ft. high? 5. Find the weight of a rectangular block of stone at 135 lb. per cubic foot, if the length of the block is 9 ft. and the other dimensions are 2 ft. and 5 ft. 6. If a cubic foot of water weighs 1000 oz., find the edge of a cubical tank that will hold 2 T. 7. Show why the statement that a rectangular parallelopiped is equal to the product of its three dimensions is the same as the statement that its volume is equal to the product of its altitude and the area of its base. 199. The Volume of a Prism. A rectangular parallelopiped can be divided into two equal triangular prisms with the same altitude and half the base. Hence, the volume of the prism is half the volume of the parallelopiped. But the base of the parallelopiped is twice the base of the prism, therefore, the volume of a triangular prism is equal to the product of its altitude and the area of its base. 200. Since any prism can be divided into triangular prisms, as in the figure, it follows that the volume of any prism is equal to the product of its altitude and the area of its base. 201. The Volume of a Cylinder. The cylinder may be divided into a number of solids that are essentially prisms, as indicated in the figure. The sum of the bases of these prisms is the base of the cylinder and the altitude of the prisms is the same as the altitude of the cylinder. Therefore, the volume of a cylinder is the product of its altitude and the area of its base. V = α × πr2. EXERCISE 33 1. Find the volume of a prism with square ends, each side measuring 1 ft. 8 in., and the height being 12 ft. 2. Find the volume of a prism whose ends are equilateral triangles, each side measuring 11 in. and the height being 20 in. 3. Find the volume of a cylinder if the diameter of its base is 20 in. and the altitude is 30 in. 4. How many cubic yards of earth must be removed in digging a well 45 ft. deep and 3 ft. in diameter ? 5. A cubic foot of copper is to be drawn into a wire of an inch in diameter. Find the length of the wire. 6. How many revolutions of a roller 3 ft. in length and 2 ft. in diameter will be required in rolling a lawn & of an acre in extent. 7. Show how to find the surface of a cylinder by dividing it into figures that are essentially parallelograms. Show how to find the surface of a prism. E F B 202. The Volume of a Pyramid. Let AB be a cube and F the middle point of the cube, then by connecting F with B, C, D and E a pyramid with a square base is formed. It is evident that by drawing lines from F to each of the ver- A tices, the cube will consist of six such pyramids. Hence, the volume of the pyramid is of the volume of the cube. The volume of the cube is the product of its altitude and the area of its base BCDE. the volume of the pyramid is of the product of the altitude of the cube and the area of its base. But the base D Therefore, |