342. A cylinder is a round body with circular ends. The line EF is called the axis or altitude, and the circular surface the convex surface of the cylinder. E 343. To find the convex surface of a cylinder, Multiply the circumference of the base by the altitude, and the product will be the convex surface (Bk. VIII. Prop. I). EXAMPLES. 1. What is the convex surface of a cylinder, the diameter of whose base is 20 and the altitude 40? We first multiply 3.1416 by the diameter, which gives the circumference of the base. Then multiplying by the altitude, we obtain the convex surface. OPERATION. 20 62.8320 40 Ans. 2513.2800 2. What is the convex surface of a cylinder whose altitude is 28 feet and the circumference of its base 8 feet 4 inches? 3. What is the convex surface of a cylinder, the diameter of whose base is 15 inches and altitude 5 feet? 4. What is the convex surface of a cylinder, the diameter of whose base is 40 and altitude 50 feet? 344. To find the solidity of a cylinder, Multiply the area of the base by the altitude: the product will be the solid contents (Bk. VIII. Prop. II). QUEST-342. What is a cylinder? What is the axis or altitude? What is the convex surface? 348. How do you find the convex surface 844. How do you find the solidity of a cylinder ? EXAMPLES. 1. Required the solidity of a cylinder of which the altitude is 11 feet, and the diameter of the base 16 feet. OPERATION. We first find the area of the base, and then multiply by the altitude: the product is the solidity. area base 16-256 .7854 201.0624 11 2111.6864 2. What is the solidity of a cylinder, the diameter of whose base is 40 and the altitude 29? 3. What is the solidity of a cylinder, the diameter of whose base is 24 and the altitude 30? 4. What is the solidity of a cylinder, the diameter of whose base is 32 and altitude 12? 5. What is the solidity of a cylinder, the diameter of whose base is 25 and altitude 15? 345. A pyramid is a solid formed by several triangular planes united at the same point S, and terminating in the different sides of a plane figure, as ABCDE. The altitude of the pyramid is the line SO, drawn perpendicular to the base. 346. To find the solidity of a pyramid, Multiply the area of the base by the altitude, and divide the product by 3 (Bk. VII. Prop. XVII). QUEST.-345. What is a pyramid? What is the altitude of a pyramid? 346. How do you find the solidity of a pyramid} EXAMPLES. 1. Required the solidity of a pyramid, of which the area of the base is 86 and the altitude 24. We simply multiply the area of the base 86, by the altitude 24, and then divide the product by 3. OPERATION. 86 24 344 172 3)2064 Ans. 688 2. What is the solidity of a pyramid, the area of whose base is 365 and the altitude 36 ? 3. What is the solidity of a pyramid, the area of whose base is 207 and altitude 36 ? 4. What is the solidity of a pyramid, the area of whose base is 562 and altitude 30? 5. What are the solid contents of a pyramid, the area of whose base is 540 and altitude 32 ? 6. A pyramid has a rectangular base, the sides of which are 50 and 24; the altitude of the pyramid is 36: what are its solid contents ? 7. A pyramid with a square base, of which each side is 15, has an altitude of 24: what are its solid contents? 347. A cone is a round body with a circular base, and tapering to a point called the vertex. The point C is the vertex, and the line CB is called the axis or altitude. C B QUEST.-347. What is a cone? What is the vertex? What is the axis? 348. How do you find the solidity of a cone ? 348. To find the Multiply the area solidity of a cone, of the base by the altitude, and divide the product by 3; or, multiply the area of the base by onethird of the altitude. (Bk. VIII. Prop. V.) EXAMPLES. 1. Required the solidity of a cone, the diameter of whose base is 6 and the altitude 11. We first square the diameter, and multiply it by .7854, which gives the area of the base. We next multiply by the altitude, and then divide the product by 3. OPERATION. 11 36.7854=28.2744 3)311.0184 Ans. 103.6728 2. What is the solidity of a cone, the diameter of whose base is 36 and the altitude 27? 3. What are the solid contents of a cone, the diameter of whose base is 35 and the altitude 27? 4. What is the solidity of a cone, whose altitude is 27 feet and the diameter of the base 20 feet? RIGHT ANGLED TRIANGLE. 349. The properties of the right angled are so important as to be worthy of particular notice. In every right angled triangle, the square described on the hypothenuse, is equal to the sum of the squares described on the other two sides. Thus, if ABC be a right angled triangle, right angled at C, then will the square D described on AB be equal to the sum of the squares E and F, described on the sides CB and A AC. This is called the carpenter's theorem. Hence, to find the hypothenuse when the base and perpendicular are known, 1st. Square each side separately. 2d. Add the squares together. 3d. Extract the square root of the sum, and the result will be the hypothenuse of the triangle. EXAMPLES. 1. The wall of a building, on the brink of a river is 120 feet high, and the breadth of the river 70 yards: what is the length of a line which would reach from the top of the wall to the opposite edge of the river? 2. The side roofs of a house of which the eaves are of the same height, form a right angle with each other at the top. Now, the length of the rafters on one side is 10 feet, and on the other 14 feet: what is the breadth of the house? 3. What would be the width of the house, in the last example, if the rafters on each side were 10 feet? 350. When the hypothenuse and one side of a right angled triangle are known, to find the other side. Square the hypothenuse and also the other given side, and take their difference: extract the square root of their difference, and the result will be the required side. 1. The height of a precipice on the brink of a river is 103 feet, and a line of 320 feet in length will just reach from the top of it to the opposite bank: required the breadth of the river. 2. The hypothenuse of a triangle is 53 yards, and the perpendicular 45 yards: what is the base? 3. A ladder 60 feet in length, will reach to a window 40 feet from the ground on one side of the street, and by turning it over to the other side, it will reach a window 50 feet from the ground: required the breadth of the street. QUEST.-349. What is the property of a right angled triangle? When can you find the hypothenuse? How? 350. How do you find a side? |