To find the diameter when the circumference is given. RULE. 1. The circumference of a wheel is 18.8496 feet; what is its diameter? 18.8496 = 3.1416 = 6 feet, diameter. 2. The circumference of the earth is about 25000 miles; what is its diameter? 3. The circumference of a tree is 93 feet; what is its diameter? 4. What is the length of a string used to draw a circumference of 37.6992 inches? To find the area of a circle, the diameter or the circumference being given. RULES. Multiply the square of the diameter by .7854; or, the square of the circumference by .07958. 1. The diameter of a pond is 50 feet; what is the area? 50 x 50 x .7854 = 1963.5 square feet, area. 2. The circumference of a race-track is one mile; what is the area? 3. The diameter of a circular park is mile; what is the area ? 4. The diameter of a circle is 400 feet, and the circumference 1256.64 feet; find the area from each dimension. 5. How many acres in a circular field whose radius is 1 mile? To find the diameter or the circumference of a circle, the area being given. RULES. Divide the area by .7854, and the square root of the quotient will be the diameter; or, divide the area by.07958, and the square root of the quotient will be the circumference. 1. The area of a circular pond is 1963.5 feet; what is the diameter? 1963.5 = .7854= 2500. V 2500= 50 feet, diameter. 2. The area of a circular field is 1 acre; what is the circumference? 3. What is the radius of a circle whose area is one square mile? 4. The area of a circle is 3 acres, 42 rods; what is the diameter? 5. The length of a fence is required that will exactly surround a circle containing one-half a square mile. 6. The area of a circle is 7854 square yards; required the diameter and the circumference. To find the side of an inscribed square, the diameter or the circumference of the circle being known. Note.— A square is inscribed in a circle when its diagonal is one of the diameters of the circle. The circle is then said to be circumscribed about the square. RULES. Multiply the given diameter by .7071; or, the given circumference by .22507. 1. The diameter of a log of walnut is three feet; how large a square stick can be hewn from it? 3*.7071 = 2.1213 feet square. 2. The circumference of a tree, exclusive of the bark, is 22.507 ft.; what is the size of the largest piece of square timber that can be cut from it? 3. What was the diameter of a log which measures 7.071 ft. across the corners after it has been squared ? 4. The side of a square field is 50 yd.; what is the circumference of the circumscribed circle? 5. The area of a circular field is one acre; what is the side of the inscribed square? REVIEW PROBLEMS. 1. What is the length in rods of the side of a square whose area is 1 acre? ANALYSIS.—As the area is the product of two equal factors, we change 1 acre to rods, and extract the square root of the number thus obtained. 1 A.=160 rods. V160=12.649 + rods. Note. As the area of any surface is the product of two factors, it is only necessary to divide the area by either factor in order to find the other. 2. What is the length of a field containing 9 acres, the form being that of a rectangle, and the width 16 rods? 3. The base of a triangular field is 20 rods, and the area 5 acres; what is the altitude of the triangle? 4. A field in the form of a rectangle is divided into two triangles by a diagonal line which measures 10 rods; the sides of the rectangle are 6 and 8 rods, respectively; what is the area of one of the triangles ? 5. The circumference of a wheel is 3.1416 yards; what is the distance from the centre to any point in the circumference? 6. One side of a field in the form of a parallelogram is 20 rods, and the area is 10 acres; what is the other side? 79. The length of a line by which a circle is inclosed is 75 yards; what is the length of a line inclosing a circle which contains 4 times the area of the first? 8. A farmer has two fields, each containing 1 acre. The first is in the form of a rectangle, one side of which is 40 yards; the second is a square field. What is the length of the fence which surrounds each field? 9. What is the value of a farm in the form of a trapezoid, the parallel sides of which measure 80 and 31 chains respectively, the perpendicular distance between these sides being 120 rods; the land being worth $15 an acre? 10. A circular piece of ground has a square laid off within it, the area of which is 1 acre; what is the diameter of the circle, if the corners of the square touch the circumference? 11. How much farther will a horse have to run in going round the sides of a square mile of land, than in going round the same area in the form of a circle? 12. Two pieces of land measure each 1 mile around. What is the difference in area, if one piece is a square, and the other a rectangle, one side of which is 100 rods in length? 13. A ladder 60 feet long is placed against a house which is 50 feet in height; the foot of the ladder rests on the ground 36 feet from the house; how far is the top of the ladder from the top of the house? 14. What is the difference in area between a circle whose diameter is one rod, and a square whose diagonal is one rod? SOLIDS. A Prism is a solid, two of whose faces are equal polygons lying parallel to each other, and the remaining faces parallelograms. The parallel polygons are called the Bases of C the prism; the parallelograms taken together constitute the Convex Surface. Prisms are named from the form of their bases; thus, a prism whose base is a triangle ---- F is called a Triangular Prism. A Right Prism is one whose edges are perpendicular to the planes of its Triangular Prism. bases. A Cube is a prism whose faces are all equal squares. A Parallelopipedon is a prism whose bases are parallelograms. A Cylinder is a round body of Cube. uniform diameter, with equal circular bases, parallel to each other. The Altitude of a prism or cylinder is the perpendicular distance between its bases. Parallelopipedon. The Convex Surface of a cylinder is the whole curved surface. The Whole Surface includes the convex surface and the surface of the two bases. Similar Solids are those which have the same number of similar faces similarly Cylinder. situated. E |