PLANE FIGURES 567. A plane surface is a surface such that if any two points in it are connected by a straight line, the straight line will lie wholly in the surface; e.g. a table top, the surface of a window pane. Test these and other surfaces by a thread held taut. 568. A portion of a plane surface bounded by lines is a plane figure ; ́e.g. a square, a triangle, a circle. 569. A plane figure bounded by straight lines is a polygon. A polygon of three sides is called what? A polygon of four sides? 570. A polygon of five sides is a pentagon; of six sides a hexagon; of seven sides, a heptagon ; of eight sides, an octagon. AREAS OF REGULAR POLYGONS 571. A polygon whose sides are equal and whose angles are equal is a regular polygon; e.g. 572. The area of any regular polygon may be found by dividing the polygon into as many equal triangles as the polygon has sides, and multiplying the area of one triangle by the number of triangles; e.g. The area of this regular hexagon is six times the area of one of the triangles, or six times one half of the product of a and b. Alt. AREAS OF TRAPEZOIDS 573. A quadrilateral having two and only two sides parallel is a trapezoid. In each of the above figures, how does the part A compare with the part B? How does the area of the trapezoid compare with that of the parallelogram which is made from the trapezoid? How is the area of the parallelogram found? Observe that in each figure the base of the parallelogram is equal to one half of the sum of the parallel sides of the trapezoid. Summary The area of a trapezoid is equal to one half of the sum of the parallel sides multiplied by the altitude. 574. Written 1. Draw a trapezoid whose altitude is 13 inches and whose parallel sides are 17 inches and 19 inches. Find its area. 2. Find the area of a trapezoid whose parallel sides are 20 feet and 25 feet, and whose altitude is 15 feet. 3. A field in the form of a trapezoid has two parallel sides of 30 rods and 35 rods; the distance between them is 20 rods. How many acres of land does the field contain? 4. A board is 1 inch thick, 12 feet long, 11 inches wide at one end and a foot wide at the other end. How many board feet does it contain? 5. A vineyard in France is in the form of a trapezoid, of which the two parallel sides are 185 meters and 155 meters, and the altitude is 130 meters. a. It has an area of how many ares? b. How many hectares? 6. Find the area of trapezoid ABCD. B A 42' C 7. The parallel sides of a trapezoid are 41 cm. and 55 cm. Its area is 1296 sq. cm. What is its altitude? Let x = the altitude. 8. The area of a trapezoid is 560.5 sq. ft. The altitude is 19 ft. The difference of the parallel sides is 5 ft. a. Find the sum of the parallel sides. b. Find the length of each of the parallel sides. STUDY OF THE CIRCLE 575. A plane figure bounded by a curved line, all points of which are equally distant from a point within, called the center, is Circumference Center Radius Diameter a circle. 576. The boundary line of a circle is the circumference. 577. A straight line passing through the center of a circle and terminating in the circumference is the diameter. 578. A straight line drawn from the cen ter to the circumference of a circle is its radius. 579. It is proved, by geometry, that the circumference of every circle is 3.1416 times its diameter. 580. Oral 1. The radius of a circle is what part of its diameter ? 2. What is the radius of a circle whose diameter is 80 cm. ? 3. What is the diameter of a circle whose radius is 35 cm.? 4. What is the circumference of a circle whose diameter is 1 foot? 5. What is the circumference of a circle whose diameter is 100 inches? 6. What is the circumference of a circle whose radius is 5 inches? 7. What is the diameter of a circle whose circumference is 31.416 inches? 8. What is the radius of a circle whose circumference is 3.1416 meters? Written 1. What is the circumference of a circle whose diameter is 50 inches? 2. What is the radius of a circle whose circumference is 182.2128 feet? 3. What is the diameter of a circle whose circumference is 7854 miles? 4. The radius of the earth is approximately 4000 miles. What is its approximate circumference? 5. The diameter of my bicycle wheels is 28 inches. a. How many feet will I travel during 700 rotations of a wheel? b. How many meters will I travel? c. How many rotations will a wheel make in traveling 1 mile? 6. A horse is tethered to a stake by a rope 50 ft. long. What is the circumference of the circle over which he can graze? FIG. 1 Observe that Fig. ABCD is a parallelogram. Its base is what of the circle? . The triangles of the circle are what part of the parallelogram? How may we find the area of the parallelogram? Of the circle? wwwww FIG. 2 Summary The area of a circle is equal to one half of the product of its circumference by its radius. By geometry it is proved also that the area of a circle is equal to .7854 of the square of its diameter, or 3.1416 times the square of its radius. How may we find the area of a circle when the radius is given? when the diameter is given? when the circumference is given? 582. Written In examples 1-12 find the area of a circle from the term given, letting D, R, and C stand for diameter, radius, and circumference, respectively: |