plier set opposite the polygon of the same number of sides, and the product will be the area. Ex. 1. Required the area of a regular octagon whose side is 20 feet. Hence, 4.8284271×400=1931.3708400, Ans. Ex. 2. What is the area of a regular decagon whose side is 87 feet? Ans. 58237.46. Ex. 3. The side of an undecagon is 40: what is its area? Ans. 14985.024. Ex. 4. Required the area of a nonagon whose side is 50 feet. Ans. 15454.56. Ex. 5. What is the area of a hexagon whose side is 25 feet? PROBLEM VI. Ans. 1623.8. To find the area of a long Irregular Figure, bounded on one side by a straight line. ART. 14. Rule.-I. Measure the D breadths in several places, and at equal distances from each other. A C B II. Add together all the different breadths, and half the sum of the two extremes. III. Multiply this sum by the base line, and divide the product by the number of equal parts of the base. Ex. 1. The breadths of an irregular figure, at five equidistant places, being 8.2, 7.4, 9.2, 10.2, 8.6, and the whole length 39, required the area. Ex. 2. The length of an irregular figure being 84, and the breadths at six equidistant places, 17.4, 20.6, 14.2, 16.5 20.1, 24.4, what is the area? Ans. 1550.64. The length of an irregular figure being 37.6, and the breadths, at nine equidistant places, 0.4.4, 6.5, 7.6, 5.4, 8, 5.2, 6.5 and 6.1, what is the area? Ans. 219.255. Ex. 4. The length of an irregular field is 50 yards, and its breadths, at seven equidistant places, 5.5, 6.2, 7.3, 6, 7.5, 7, and 8.8 yards, what is its area? Ans. 349,916 sq. yds. NOTE. If the perpendiculars or breadths be not at equal distances add them together, and divide their sum by the number of them, for the mean breadth; then multiply the mean breadth by the length, and the product will be the whole area not far from the truth. PROMISCUOUS EXAMPLES. 1. What is the area of an equilateral triangle whose side is 20 rods? Ans. 2. Required the number of square yards in an equilateral triangle, whose side is 10 yards. Ans. 43.3. 3. What is the area of a triangle whose base is 18 feet 4 inches, and height 11 feet 10 inches? Ans. 108.5 in. A 4. How many square yards are there in a trapezium, whose diagonal is 48 feet, and whose perpendiculars are 16 and 14 feet? Ans. 5. How many square rods in a trapezoid, whose parallel sides are 38 and 26 rods, and whose breadth or height is 18 rods? Ans. 6. Required the area of an octagon, whose side is 22 feet, and the perpendicular, from the centre on one of the sides, 12.478 feet. Ans. 7. What is the area of a pentagon, whose side is 8 feet 4 inches, and the perpendicular, from the centre on one of the sides, 4 feet ? Ans. 8. If the length of an irregular figure be 43.5, and the breadths, at six equidistant places, be 4, 6.5, 6, 7.5, 8, and 8.5, what is the area? Ans. 9. There is a triangular lot of land whose base, or longest side, is 51 rods, and the perpendicular, from the opposite corner to the base, measures 44 rods; how many acres does it contain ? Ans. 10. The length of an irregular piece of land being 21 chains, and the breadths, at six equidistant points, being 4.35, 5.15, 3.55, 4.12, 5.02, and 6.10 chains: required the Ans. area. 2. A Diameter of a circle is a straight line, passing through the centre and terminating at the circumference; as AB (fig. 1.) 3. A Radius or Semi-Diameter is a straight line, extending from the centre to the circumference; as CA or CD (fig. 1.) 4. A Semi-circle is one half of the circumference; as ADB (fig. 1.) 5. A Quadrant is one quarter of the circumference; as, AB (fig. 2.) A B2 6. An Arc is any portion of the circumference; as AC B (fig. 3.) 3 C 7. A Chord is a straight line, which joins the two extremes of an arc; thus, AB is a chord of the arc ACB, (fig. 3.) A B 8. A Circular Segment is the space contained between an arc and its chord; as, ACB (fig. 3.) The chord is sometimes called the base of the segment. The height of the segment is the perpendicular from the middle of the base to the arc. 9. A Circular Sector is the space contained between an arc and the two radii, drawn from the extremes of the arc: thus, CAB (fig. 4) is a sector. 10. A Circular Zone is the space contained between two parallel chords which form its bases; thus, ABCD (fig. 5.) It is called the middle zone when the chords are of equal length. 11. A Circular Ring is the space between the circumferences of two concentric circles; thus, AB and DE (fig. 6) is a A circular ring, having a common centre, C. |