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1. What is the area of a pentagon, whose side is 95, and its apothegm 65.36 ?
2. What is the side of a pentagon whose area is 1?
Ans. 0.7623, nearly. Solution, T.720477 0.5812 the square of the side of a pentagon whose area is 1, and V0.5812.7623, the required side.
To find the area of a pentagon by the sliding rule :-Place 1 on Cover .7623 on the line D, and over the side of any pentagon found on D, will be found its area on the line C; or, if the side of a pentagon be given in inches and decimal parts of an inch, and its area is required in feet, (because 9.15 inches is the side of a pentagon which contains one square foot,) place 1 on C over 9.15 on D, and over the side in inches found on D, will be found the area in feet on the line C.
3. How many square feet in a pentagon 10 inches on a side? in one 14 inches on a side? in one 20 inches on a side? in one 25 inches on a side? in one 30 inches on a side? in one 35 inches on a side? and in one 40 inches on a side? Answers in order,—1.18; 2.51; 4.75; 7.45; 10.7; 14.62; 19.16 square
4. What is the side of a hexagon whose area is 1?
Ans. 0.621, nearly.
Solution, 7.53907 the square of the required side, the square root of which is 0.621, the required side.
5. What is the side of a hexagon whose area is one acre, or 160 square rods? Ans. 7.861 rods.
6. How many acres in a hexagon whose side is 12 rods? in one whose side is 20 rods? and in one whose side is 60 rods?
Solution by the sliding rule :-Place 1 on C over 7.86 on D, and over 12 rods on D we find 2.32 acres; and over 20 we find 6.45 acres; and over 60 we find 58.45 acres.
7. What is the side of an octagon whose area is 1?
8. What is the side of an octagon whose area is 144 square inches? Ans. 5.46 inches. 9. What is the side of an octagon whose area is one acre? Ans. 5.76 rods, nearly.
10. How many square feet in an octagon whose side is 6 inches? in one whose side is 10 inches? in one whose side is 15 inches? and in one whose side is 20 inches?
Solution.-Place 1 on C over 5.46 on D, and over the side in inches found on D, will be found the areas in feet on the line C. Answers in order,-1.21; 3.33; 7.52; 13.25 feet.
11. How many acres in an octagon, whose side is 5.9 rods? in one whose side is 8 rods? in one whose side is 15 rods? in one whose side is 30 rods? and in one whose side is 40 rods? Answers in order,-1.047 acres; 1.925 acres; 6.76 acres ; 27.1 acres; and 48 acres.
12. How many square yards in a decagon, whose side is 12 feet? Ans. 123.1072 yards.
T 22. CIRCLE.
The circle is a plane figure bounded by a curved line, every point or part of which is equally distant from a certain point within called the centre. The circumference, or bounding line, is sometimes called a circle.
By examining the figures representing the regular polygons, it will be seen, that they may be divided into as many triangles as the polygon has sides; and that, consequently, their areas may be found by multiplying the perimeter of the polygon by half the apothegm, or half the radius of the greatest inscribed circle. Now if a polygon of a great number of sides be drawn in a circle, its perimeter will evidently approach very near to
the circumference of the circle; and if we should suppose the number of sides of the inscribed polygon to be absolutely infinite, its perimeter would evidently coincide with the circumference of the circle: the circumference of the circle may, therefore, be considered the perimeter of a polygon of an infinite number of sides; and hence it may be regarded as the sum of the bases of an infinite number of triangles, whose vertexes all meet at the centre of the circle. Therefore
To find the area of a circle:
Multiply the circumference by half the radius, or one-fourth of the diameter.
The exact ratio of the circumference of the circle to its diameter, though for a long time a celebrated problem with geometricians, never has been and never can be exactly obtained; yet we may easily find the ratio of the diameter of a circle to its circumference sufficiently near for any practical or scientific purpose.
There are various methods of approximating towards the circumference of the circle. The method of approximation adopted by the ancients was as follows. They supposed a polygon of a great number of sides to be inscribed in the circle, and a polygon of the same number of sides to be circumscribed around the circle; they then, by a plain but laborious method, calculated the perimeters of the inscribed and circumscribed polygons, and took the arithmetical mean between them for the true circumference. By this method, the indefatigable Ludolphus Van Ceulen, a Dutch mathematician, (who died in 1610,) calculated the ratio of the diameter of a circle to its circumference, true to thirty-five decimal places. Thus, calling the diameter 1, he found the circumference to be 3.14159265358979323846264338327950288.
That prince of mathematicians, Sir Isaac Newton, discovered a much more rapid method of approximation, which we will give for the gratification of those who are curious in such
As we have shown, under the hexagon, 21, the chord of 60° (that is, one side of the hexagon) equals radius, (or the
semidiameter of the circle,) hence the sine of 300 (that is, half the chord of 60°) equals half of radius; and if ab, the sine of 300, be represented by x, and radius by r, then ac, the arc of 3.x5 5x7 35x9 +
30°, will = x + + +
67.2 404 1127.6 11528 281610
dius, or r, = 1, then ab, the sine of 30°, represented in the series by x, will be ; and therefore the series may be ex
6435 73014444032' stand thus:
&c., which, expressed in decimal form, will
The arc ac = 0.523593775598296
Since 300 is of 360°, the arc of 300 will be of the circumference of the circle of which it is a part; and since we have calculated the are of a circle whose diameter is 2, if we multiply the said arc by 6, it will give the circumference of a circle whose diameter is 1. Thus: