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Although the length of the circumference of a given circle, or its ratio to the diameter, has never been exactly obtained ; yet it may be approximated to any required degree of exactness. \ 189. Problem. To find the approximate ratio of the circumference to the diameter of the circle. Designating. this ratio by ar, the radius of any circle by R, the circumference of a circle whose radius is R., by C, and the

o T area of the same circle, by A., we have t = I T 3 T : * * *

whence C = 2 or X R, and A = 2 at X R × 4 R =
T. Jo 2. -
If R = 1, then A = or ; that is—The area of a circle
whose radius is one, equals the value of it ; that is, the
area of the circle whose radius is 1, bears the same ratio
to the square whose side is one, as the circumference bears
to the diameter. The question is therefore reduced to
finding the area of a circle whose radius is unity or one.
The area of a circle being less than that of any circum-
scribed polygon and greater than that of any inscribed ;
and as the values of two similar polygons, the one in-
scribed and the other circumscribed, approach more
nearly to an equality, in proportion as the number of their
sides is greater ; and as the number of sides of these poly-
gons may be increased till the difference in their areas
shall be less than any assignable quantity d : If then
we take the arithmetical mean between the areas of the
circumscribed and inscribed polygons as the area of the
circle, the error will be much less than half d.
A square circumscribed about a circle whose radius is
1, is the square of the diameter, or the square of 2, equal
to 4 ; the inscribed square (143) is the square of

(2)}, or 2. The proposed approximation, therefore, will be readily made if we can find an easy solution of the following problem. 190. Problem. The area of a regular inscribed polygon and that of a similar circumscribed polygon being given, to find the areas of regular inscribed and circumscribed polygons of double the number of sides. Let EF and GH (fig. 106) be the sides of regular polygons of n sides, the Fig.106. one circumscribed and the other inscribed ; by drawing the chords BG and BH, and the tangents GI and HK, the straight lines BG and IK will be the sides of regular

polygons of 2n sides, the one inscribed and the other circumscribed. The triangles ECF and GCH will be contained n times in the respective circumscribed and inscribed polygous of n sides, of which they are parts; and the triangles ICK and BCG, will be contained 2n times, in the regular circumscribed and inscribed polygons of 2n sides. Let A represent the area of the regular circumscribed polygon of n sides ; a the area of the similar inscribed polygon ; A' and a' the areas of the regular circumscribed and inscribed polygons of 2n sides. We shall have the areas of the several polygons as follows: circumscribed, of n sides = A = 2 m × EBC, inscribed, of n sides = a = 2 m × GDC, circumscribed, of 2n sides = A' = 4 m × BCI, inscribed, of 2 m sides = a' = 2 m × GBC. T} A EBC a' GBC len a = GBG, and a = GDG : the same altitude being to each other as their bases, we EBC CE B . GBC T CG CD ' h GBC CB EBC * GDC T CD GBC I gives a' = (A X a)*. to- - o (1). Also, because CI bisecting the angle BCE, divides the base into parts BI and IE, proportional to the sides CB and CE, we have BCI BI BC st BCI BCI

and triangles of

have for the same reason, we

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a --- as The problem then is solved by means of these two formulas; for knowing the areas of the inscribed and circumscribed polygons of n sides, we have the quantities A and a ; from these, the formulas give us A' and a', the areas of the inscribed and circumscribed polygons of 2 n sides. 191. We are now prepared to solve the problem enunciated in article 189, viz. –To find the area of a circle whose radius is one ; or, in other words, to find the value of it. The area of the circumscribed square, radius being

1, is equal to 4; the area of the inscribed square is equal to 2. Making A = 4, and a = 2, the formulas (1), (2), 1 will give a' = (S) = 2,82S4271 = the area of an in- 16 6 so grassfär-881970s; = the area of a circumscribed octagon. If we substitute these values of A' and a', for A and a in the formulas, we shall have a' = 3,061.4674 = the area of a regular inscribed polygon of 16 sides, and A' = 3, 1825.979 = the area of a regular circumscribed polygon of 16 sides. The values of these polygons will enable us to find the areas of inscribed and circumscribed polygons of 32 sides; and, as the farther we proceed in the calculation the nearer will the two polygons approach to an equality, we may continue the process till the values of the inscribed and circumscribed polygons do not differ for any number of decimals to which it may be thought best that the expressions should extend. Having carried the process thus far, and knowing that the circle cannot be greater than the circumscribed polygon, nor less than the inscribed polygon, we take this value for the area of the circle, as far as the expression extends. The following table gives the result of this calculation pursued till the expressed values of the circumscribed and inscribed polygons in a circle whose radius is 1, do not differ for the first seven decimals.

scribed octagon; and A' =

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Fig.107.

We therefore take 3,1415926 for the approximate area of a circle whose radius is 1. It is also the approximate value of the ratio of the circumference of any circle to its diameter, that is, T = 3, 1415926 nearly ; and multiplying the diameter of a circle by this quantity, will always give us the circumference to a sufficient degree of eXaCtneSS,

192. If we denote the radius of a circle by R, the diameter will be 2 R, and the circumference 2 or . R, and the area, A = } R × 23r. R = it. It?. That is—The square of radius multiplied by 3,1415926, will give the area of any circle.

193. Let A denote the area of a circle whose radius is It, and a the area of a circle whose radius is r ; we have the two equations, A = gr. R2, a = T. r2. These two

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ing each term of the second ratio by the common factor

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e © . . . e. . A equations will give the proportion - Co.

to each other as the squares of their radii. The radii of
circles are in the ratio of their diameters, and therefore
in the ratio of their circumferences (148). We see,
therefore, that it is with curvilinear as with rectilinear
figures (168); and we have this general rule—Similar
plane figures are to each other as the squares of their ho-
mologous lines.
194. The sector AEBC (fig. 107) is evidently the
same part of the circle that the arc AEB is of the entire
circumference ; if the circumference multiplied by half
radius gives the area of the entire circle, it follows that
—The arc of the sector multiplied by half radius, gives
the area of the sector.
To obtain the area of the segment AEBD, subtract
the area of the triangle ABC from the area of the sector.

PROBLEMIS.

(1). Suppose a castle, whose walls are 48 feet high, surrounded by a ditch 64 feet wide; what is the length of a ladder which will reach from the outside of the ditch to the top of the castle wall !

(2). A triangular field has one right-angle; the side opposite the right-angle measures 75 chains; one of the sides adjacent to the right-angle, measures 45 chains; what is the length of the third side 3 (3). How many square chains in a rectangular field whose length is 12# chains, and whose width is 8 chains 1 (4). What is the side of a square field of an equal area with the above 7 (5). One side of a triangular field is 20 rods; and the perpendicular distance of the vertex of the opposite angle from this side, is 20 rods; what is the side of a square field whose area is twice as great (6). What length of carpeting of a yard wide, will cover the floor of a room whose length is 30 feet and width 21 feet 1 (7). The length of a rectangular lawn is 25 rods, and its width S rods; what is the length of an equivalent lawn whose width is 124 rods (8). What is the circumference of a circle whose radius is 4 1 and what is the area of the same circle 1 (9). What is the approximate value of the side of a square equivalent to a circle whose radius is 97 * (10). What is the radius of a circle whose circumference is 31,416 '' (11). What is the value of the apothegm of a regular hexagon, inscribed in a circle whose radius is 20 ! and what is the area of this hexagon 7 (12). What is the area of an equilateral triangle inscribed in a circle whose radius is 20 ! (13). What is the area of a square inscribed in a circle whose radius is 10 ! What is the area of the circumscribed square 1 (14). What is the area of a semi-circular pond whose straight side is 200 yards 1 (15). What is the area of a sector embracing 600 in a circle whose radius is 20 yards 1 §

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