A Rectangle (Art. 166) is a right-angled parallelogram ; a Rhomboid is a parallelogram having no right angles; and a Rhombus is a rhomboid having equal sides. 394. It will be seen by the diagram that the rhombus A B C D is equal to the rectangle EBCF of the same base and altitude (Art. 218). E Hence, A F D The area of a parallelogram is equal to the product of the base and altitude. 9. What is the area of a parallelogram whose base is 36 feet and altitude 15 feet ? 10. The base of a rhombus is 16 feet and its height 12 feet. What is its area ? 11. What is the difference in the area of two floors, the one being 37 feet long and 27 feet wide and the other 40 feet long and 20 feet wide ? 395. A Trapezoid is a quadrilateral having only two of its sides parallel. 396. The area of a trapezoid is equal A Trapezoid. to the product of half the sum of the parallel sides and the altitude. 12. What is the area of a trapezoid, the longer of the two parallel sides being 120 feet, the shorter 100 feet, and the altitude 85 feet ? 13. What is the area of a plank whose length is 6 meters, the width of one of the parallel ends being 60 centimeters and the other 40 centimeters ? 14. The parallel sides of a field are 131 and 243 yards, and the breadth 220 yards. How many acres does it contain ? 397. A Trapezium is a quadrilateral having no two of its sides parallel. A Diagonal is a straight line joining any two angles of a plane figure not adjacent, as the line A C. A Trapezium. 398. It will be seen from the above diagram that a diagonal divides a trapezium into two triangles. Hence, The area of a trapezium is equal to the product of the diagonal and half the sum of the perpendiculars drawn to the diagonal from the vertices of opposite angles. Note 1. — Any plane figure bounded by straight lines is called a Polygon, and may be divided into triangles; and the sum of the areas of the triangles will be the area of the figure. NOTE 2. --- For the Circle see Art. 221. 15. The diagonal of a trapezium is 16 feet, and the perpendiculars upon it from the opposite angles are 7 feet and 5 feet. Find the area. 7 ft. + 5 ft. 2 16. What is the area of a trapezium whose diagonal is 65 feet, and the length of the perpendiculars let fall upon it from opposite angles is 14 feet and 18 feet ? 17. How many square yards of paving are there in a trapezium whose diagonal is found to measure 126 feet 3 inches, and the perpendiculars upon it 58 feet 6 inches and 65 feet 9 inches? PRISMS. 399. A Prism is a body having two equal parallel polygons as bases and the other faces parallelograms. A prism is triangular, quadrangular, pentagonal, etc., according as its bases have three sides, four sides, five sides, etc. A Triangular Prism. Prism. 400. The contents of a prism are equal to the product of the area of the base by the altitude A Quadrangular or length. NOTE. — For the Cylinder see Art. 225. 18. What are the contents of a triangular prism whose length is 15 feet and the area of its triangular base is 6 square feet ? 19. What are the contents of a quadrangular prism whose length is 6 meters, and wbose base is 18 by 20 centimeters ? 20. The altitude of a pentagonal prism is 20 feet 6 inches, and the area of its base 1075.30 square inches. What are its contents in cubic feet? PYRAMIDS AND CONES. 401. A Pyramid is a body whose base is any polygon, and whose sides are triangles meeting at a point called the vertex of the pyramid. A pyramid, like a prism, is triangular, quadrangular, pentagonal, etc., according to the form of its base. 402. A Cone is a body whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex of the cone. The altitude of a pyramid or cone is the shortest distance from the vertex to the center of the base; as A B. The slant height is the shortest distance from the vertex to the perimeter of the base; as A C. 403. The contents of a pyramid or cone are equal to the product of its base by one third of the altitude. A Cone. 21. What are the contents of a triangular pyramid whose altitude is 14 feet 3 inches, and the area of whose base is 14.70 square feet? 22. What are the contents of a cone whose altitude is 15.06 meters, the circumference of the base being 12.5 meters ? 23. The largest of the Egyptian pyramids is square at its base, and measures 693 feet on a side. Suppose its other sides to meet at a point 500 feet above the base. What are the contents of the pyramid in cubic feet ? 404. A Frustum of a pyramid or a cone is the part which remains after cutting off the top by a plane parallel to the base. 405. The contents of the Pyramid. jrustum of a pyramid or Cone. cone are equal to the sum of the areas of the two bases plus the square root of their product, multiplied by a third of the altitude. Frustum of a Frustum of a 24. What are the contents of the frustum of a square pyramid whose altitude is 30 feet, and whose side at the base is 20 feet and at the top 10 feet? Solution. V 40000 = 200; 200 + 400 + 100 = 700. 30 ; 3= 10; 700 x 10 = 7000 cu. ft. 25. What are the contents of a column whose altitude is 28 feet 6 inches, and whose diameter at the larger end is 3 feet and at the other 2 feet 6 inches ? 26. How many cubic feet in a square stick of timber whose. length is 18 feet 8 inches, and whose side at the larger end is 27 inches and at the smaller is 16 inches ? A Sphere. 406. A Sphere is a body bounded by a curved surface, all parts of which are equally distant from a point within called the center. The Diameter of a sphere is any straight line drawn through its center, and terminating both ways in the surface; and the Circumference is the greatest distance around the sphere. 407. The surface of a sphere is equal to the product of 3.1416 by the square of the diameter ; and The contents of a sphere are equal to the product of į of 3.1416 by the cube of the diameter. 27. What is the surface of a sphere whose diameter is 25 inches ? .......... 28. What number of square meters of gold-leaf will gild a globe 18 centimeters in diameter ? |