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Ex. 6. What is the area of a hexagon, whose lide is 5*
Ans: 6495.18 square yards. Ex. 7. Required the area of a hexagon, whose side is 20 Feet.
Ans. 1039.2238 Ex. 8. Required the area of a pentagon, whose fide is 4.
Anf. 27.5276. Ex. 9. How many square yards are in a decagon, whose Ede is 12 feet?
Anf. 123 yards 11 inches 6 pts,
The area of a polygon being given to find the side.
Divide the area of the given polygon by the tabular area of the like polygon, and the square root of the quotient will be the fide of the given polygon.
Required the side of a pentagon, whose area is 61.9371.
36=6 the side of poly.
10322850 Required the side of a decagon, whose area is 3077.682. Square yards.
Anf. 20 yards. Ex. 3 What is the side of a trigon, whose area is 173.2025 square yards ?
Anf. 20. Ex. 4. Required the side of a pentagon, whose area is 27.5276 square yards.
Anf. 4 yards.
A circle may be considered as a regular polygon of an ins finite number of fides.
The area of a circle is equal to the area of a triangle, whose base is equal to the circumference, and height the radius.
The proportion of the diameter to the circumference may be found thus : Describe a polygon of a great number of fides, about a circle of a known diameter, and inscribe another of the like number of sides ; find the perimeter of each, and the square root of their product will be the circumference. Hence the circumference of a circle whose diameter is 1, is 3.141592653 58979323846264338327950288, of which number 3.1416 may be used, it being fufficiently accurate for most practical pur. poses.
PROBLEM XIV. Plate 6. fig. 82.
The diameter being given to find the circumference
Multiply 3.1416 by the given diameter; the product will be the circumference.
Required the circumference of a circle, whose diameter is
for 61 cogs.
9.4656 Ex. 2. The distance between the cogs of a mill-wheel, and the centre of the axle, is 5 feet; how many cogs, at 4 inches pitch, will the wheel admit of ?
Anf. 94.248 cogs. Ex. 3. The same thing being given, required the true pitch,
Anf. 6 inches 2" 2" nearly. Ex. 4. Supposing the earth to be an exact sphere, required its circumference, the diameter being 7958 miles.
Anf. 25000.8528 miles. Ex. 5. Required the circumference of a circle, whose radius is 15 feet.
Anf. 9412486 feet. Ex 6. What is the circumference of a circle, whose diame. ter iz 48 inches ?
Anfo 12 feet 8 inches 41 parts.
The circumference of a circle being given, to find the diameter.
Divide the circumference by 3.1416, and the quotient will be the diameter.
EXAMPLE I. Required the diameter of a circle whose circumference is 9448.
Ex. 2. What is the diameter of a circle whose circumference is 80 feet?
Anf. 25.464 feet. Ex. 3. Required the diameter of a circle, whose circumference is 1000.
Anf. 318309. Ex. 4. Required the diameter of a circle, whose circum. ference is 64 feet.
Anf. 20.368. Ex. 5. Required the diameter of a milftone, whose circumference is 22 feet.
The diameter and sircumference of a circle, being given, to find
Multiply half the radius into the circumference, and the product is the area.
Required the area of a circle whose diameter is in
.25 half the rada
CIRCLES are similar figures.; and similar surfaces are as the squares of their corresponding fides; therefore, as 1 ° is ta. 7854 : (the area of a circle whose diameter is 1) so is the square of any given diameter to the area required. Hence,
Multiply the square of the diameter by the common number, :7854, and the product is the area.
Required the area of a circle, whose diameter is 12 feet.
Anf 113.0976 Square feet. Ex. many square feet are in a table, whose diameter, is 6 feet?
Anf. 28.2744 feet. Ex. 4. What is the area of a circular court, whose diameter is 24. yards ?
Ans: 452.3904 Square yards. Ex. 5. How many fquare miles are in a great circle of the carth, its diameter being 79574 miles ?
Anf. 49736071.58880750. Ex. 6. What is the arca of a circle, whose diameter is 3k feet?
Anf. 11.0446875 square feet. Ex. 7. Required the area of a circular garden, whose radius is 160 links of the English chain. Ang 3 roods & poles 20 yds. Ex. 8. What is the area of a circle, whose diameter is 2 feet?
Anf. 3.1416. Ex. 9. Required the area of the ring between the circumferences of two concentric circles, their diameters being 20 and
Anf. 137.445 sq. inches.