b. Exercises. (1) A baseball diamond is a square 90 feet on a side. Find the area. (2) One parallel side of a trapezoid is 15 feet, and its altitude is 6 feet. If the area is 75 square feet, find the other parallel side. 10 ft. Answer. (3) The diagonal of a rectangle is 37 units. Find the area if one side is 12 units. (4) Find the side of a square whose area is equal to the sum of the areas of three squares whose sides are 8, 9, and 12 units. 17 units. Answer. (5) Find the area of a trapezoid with altitude 6 and bases 7 and 13. (6) Find the altitude of a parallelogram with a 12-foot base, if the area is 120 square feet. 10 ft. Answer 58. Special properties of miscellaneous figures.-a. An ellipse is the path of a point whose distance from two fixed points is a constant The orbits of the earth and other planets are ellipses. Ellipses are also used in making machine gears. sum. FIGURE 100.-Ellipse. b. A parabola is the path of a point which is equidistant from a line and a fixed point not on the line. The parabola has the property that if a light is placed at its fixed point, all of the light is reflected in parallal rays. Therefore parabola reflectors are used for searchlights. The reverse process is also true and therefore reflecting telescopes use parabolic mirrors. The paths of projectiles in a vacuum are parabolas. Also the graphs of many scientific formulas are parabolas. FIGURE 101.-Parabola. c. A hyperbola is the path of a point whose distances from two fixed points have a constant difference. ж FIGURE 102.-Hyperbola. Hyperbolas were used during World War I in sound ranging to locate distant invisible enemy guns. d. The ellipse, parabola, and hyperbola each have an equation and the curves may be obtained by graphing the equation. This was done in the section on graphing for the parabola. 59. Solid geometry-Definitions and properties of some geometric solids.-a. Polyhedron.-A geometrical solid formed by portions of planes called faces, whose lines of intersection are called edges and whose points of intersection are called vertices. (1) Prism.—A polyhedron generated by a plane polygon moving through space always parallel to a fixed plane and in such a way that the vertices of the polygon move along straight lines. The bases are (a) A FIGURE 103.-Triangular (a) and pentagonal (b) prisms. (b) congruent polygons, and the faces are parallelograms. The volume of a prism is the product of the area of the base times the altitude (the perpendicular distance between the polygon bases). V=Bh where B represents the area of the base. Example: Find the volume of a square based pyramid with altitude 6 inches and side of base 4 inches. B=4.4=16 sq. in. (2) Pyramid.-A polyhedron which has a polygon for a base and the other faces are triangles all of which meet at a common vertex. FIGURE 104.-Pyramid. The volume of a pyramid is equal to one third the product of the area of the base by the altitude (the perpendicular distance from the common vertex to the base). 1 Example: Find the volume of a square-based pyramid whose altitude is 6 feet and whose base edge is 4 feet. b. Cylinder.—A surface generated by a closed plane curve moving along a straight line parallel to a given plane. The altitude of a FIGURE 105.-Cylinder. cylinder is the perpendicular distance between the parallel bases. The volume of a cylinder is equal to the product of the area of the base by the altitude. V=Bh The most commonly used cylinder is the circular cylinder. Example: Find the volume of a circular cylinder whose altitude is 5 inches and the radius of the base is 7 inches. c. Cone. A surface generated by all projection lines from a fixed point to a plane closed curve. The fixed point should not be in the same plane as the curve. A right circular cone is made by revolving FIGURE 106.-Right circular cone. a right triangle about one of its sides as an axis. It is the most commonly used cone. The volume of a cone is one-third the product of the base area by the altitude. V=Bh Example: Find the volume of a cone with 4 feet altitude and circular base with radius 8 feet. B=r2=3.14X8X8 =3.14X64=200.96 sq. ft. V=X200.96×4 =267.95 cu. ft. Note the similarity in the formulas for the volumes of prisms and cylinders, also pyramids and cones. d. Sphere. A closed surface whose points are all equally distant from a fixed point inside the surface called the center. FIGURE 107.-Sphere. (1) A line segment (R, fig. 107) which joins any point on the surface to the center is a radius of the sphere. (2) A line segment (D, fig. 107) which passes through the center and has its end points on the surface is a diameter of the sphere. (3) The volume of a sphere is given by the formula: 4 (4) The surface area of a sphere is given by the formula: S=4πr2 Example: Find the surface area and volume of a sphere of 4-foot radius, |