The truth of this will easily appear, by considering that if the circumference of a circle be 1, the diameter=0.31831, (Art. 18,) and of the product of this into the circumference is .07958, the area. But the areas of different circles being as the squares of their diameters, are also as the squares of their circumferences. Consequently, the area of a circle is readily found, by multiplying the square of the circumference by .07958. Ex. 1. If the circumference of a circle be 136 feet what is the area? Ex. 2. What is the area of a circle whose circumfer ence is 113? Ans. 1016.158. Ex. 3. How many square feet are there in a circle whose Ans. 86.5933. circumference is 10.9956 yards ? Ex. 4. What is the area of a circle whose circumference is 67? Ans. 357.234. Ex. 5. What is the area of a circle whose diameter is 39.34 inches? Ans. 1240.98 sq. in. Ex. 7. Required the area of the two ends of a cylinder whose diameter is 3 feet. Ans. 7.068 ft. PROBLEM V. To find the area of a Circle when the Diameter only is known. ART. 21. Rule.-Multiply the square of the diameter by the decimal .7854, and the product will be the area. Ex. 1. What is the area of a circle whose diameter is 11 feet? -2 OPERATION. 11=121×.7854-95.0334, Ans. Ex. 2. What is the area of a circle whose diameter is 8.75 feet? Ans. 60.1320. Ex. 3. Required the area of a circle whose diameter is 92.75 feet. Ans. 6756.436. PROBLEM VI. To find the diameter of a Circle when the Area only is known. ART. 22. Rule. Divide the area by the decimal .7854, and the square root of the quotient will be the diameter. This is just the reverse of the rule in Art. 19. Ex. 1. What is the diameter AB of a circle whose area is 380.1336? B OPERATION. 380.1336-.7854-484 And484-22 A Ex. 2. If a horse be tied by the head with a cord fastened to a post, so as to be able to graze exactly two acres of meadow, how long must the cord be? Ans. 10.0925 rds. Ex. 3. There is a meadow of 10 acres in the form of a square, and a horse tied equidistant from each angle or corner; what must be the length of the rope that will permit the horse to graze over every part of the meadow? Ans. 28.284+rods. Ex. 4. If the area of my garden be 95.033 square rods, what is the circumference and diameter of a circular garden of equal contents ? 34.557. circum. in rds. 11 rds. diam. Ans. PROBLEM VII. To find the length of an Arc of a Circle, when the number of degrees which it contains and the Radius are known. ART. 23. Rule I.-Multiply the number of degrees in the arc by the decimal .01745, and that product by the radius of the circle. Or, Rule II.-As 3 is to the number of degrees in the arc, so is .05236 times the radius to its length. Ex. 2. What is the length of an arc of 20 degrees, in a circle whose radius is 45 feet? STATEMENT BY RULE II. As 3:20::.05236×45:15.708, Ans. Ex. 3. What is the length of an arc containing 15 degrees and 15 minutes, the diameter of the circle being 20 yards? Ans. 5.32225. NOTE. When the arc contains degrees and minutes, as in the last example, reduce the minutes to the decimal of a degree, which is done by dividing them by 60. The length of an arc is frequently required when only the chord and the height are given, in which case the length of the arc may be found by the following approximating Rule. From 8 times the chord of half the arc, subtract the chord of the whole arc, and of the remainder will be the length of the arc, nearly. Ex. 1. What is the length of an arc whose chord is 120, and whose height is 45? PROBLEM VIII. To find the side of a Square inscribed in a Circle, from its circumference or Diameter. ART. 24. Rule.-I. Multiply the diameter by .7071 =the side of the inscribed square. II. Multiply the circumference by .2251-side of the inscribed square. NOTE. The area of a circle is to the area of the circumscribed square as .7854 is to 1, and to that of the inscribed square as .7854 is to. Consequently, the square within the circle is precisely half of the square without. A B Ex. 1. What is the side of a square inscribed in a circle whose diameter AB is 200 feet? OPERATION. .7071×200=141.4200=the side of the inscribed square. Ex. 2. What is the area of a square inscribed in a circle whose area is 159? .7854 ::: 159 : 101.22 = area. Ex. 3. What is the area of a square circumscribed about a circle whose area is 159? .7854:1:: 159: 202.44. Ex. 4. The circumference of a circular walk is 780: what is the side of an inscribed square? Ans. 175.578. Ex. 5. The circumference of a circular pond is 312 feet: what is the side of the largest square sheet of ice which can be cut from it when frozen over? Ans. 70.2312 ft. Ex. 6. The circumference of a circle is 715 : what is the side of an inscribed square? Ans. 100.9445. |