EXAMPLES.-1. To find the area of a triangle, whose sides

are 4, 5, and 6.

a+b+c 15

Here a=4, b=5, c=6, p=(・

=) 7.5. and

2

p.p-a.p-b.p-c=7.5 x 7.5-4x7.5-5x7.5-6=

√7.5 × 3.5 × 2.5 × 1.5= √✓/98.4375=9.9215—the area required. 2. Required the area of a triangle, of which the three sides are 20, 30, and 40, respectively? Ans. 290.4737, &c.

3. The sides are 12, 20, and 25, required the area of the triangle ?

285. To find the area of a regular polygon, having one side, and also the number of sides given.

RULE I. Let ABDEF be any regular polygon, bisect the angles FAB, ABD by the lines AC, BC, and from the point of intersection Clet fall the perpendicular CH.

II. Let n the number of sides of the polygon, a≈CH, and b=

AB, then will

polygon 1.

F

E

This rule is given, without a demonstration, in the Geodesia of Hero the younger; but the invention is supposed to belong to some preceding, and more profound Geometer. Tartalea is the first among the moderns who introduces the rule, viz. in his Trattato di Numeri et Misure, fol. Venice, 1556.

ba

2

This rule is evident, for the area of each of the triangles will be (Art. 283.); but there are n triangles, wherefore the area of their sum, (viz.

ba nba

of the given polygon,) will be nx 2 2

If the side of each of the following figures be unity, then will the radius of the inscribed and circumscribed circles be as below:

EXAMPLES.-1. The side of a pentagon is 4, and the perpendicular from the centre 2.01, required the area?

2. The side of a hexagon is 7.3, and the perpendicular from the centre 6.32 required the area?

3. To find the area of an octagon, whose side is 9.941, and perpendicular 12. Ans. 477.168.

4. To find the area of a heptagon, whose side is 4.845, and perpendicular 5.

Inscribed circle. Circum. cir. Perp. height.

Equilateral triangle 0.2886751310.57735027|0.86602540

Hence the areas of these figures may be readily found, and likewise those of similar figures, whatever be the length of the given side; since similar polygons are to one another as the squares of their homologous sides, (Euclid 20. 6.) or as the squares of the diameters of their circumscribing circles by 1. 12.

If the square of the side of any regular polygon in the following table, be multiplied into the number standing against its name, the product will be the

286. To find the area of any given rectilineal figure ABCDEF.

areas together, the sum will be the area of the figure ABCDEF. EXAMPLES.-1. Let AC=10, BH=4, CL=6, AD=12, CL=6, FD=8, EN=3, and FK=5.

AC.BH 10 x 4

40

Then

=20=area of ABC.

2

Their sum 98 area of ABCDEF.

2. Let AC=45, BH=10, AD=50, CL=20, FD=100. EN=20, and FK=14, to find the area. Ans, 2075.

287. The diameter of a circle being given, to find the circumference; or the circumference being given, to find the diameter.

RULES I. As

7:22

or, as Î13 : 355

or, as 1: 3.1415927

:: the diameter: the circumference.

a The first of these proportions is that of Archimedes, which is the easiest, although the least exact, of any of the rules which have been proposed for this purpose; the second proportion is that of Metius; the third is Van Ceulen's rule, and depends on Art. 252, where it is shewn, that if the diameter be 2, the circumference will be 6.2831853, &c. wherefore, if the diameter be 1, the circumference will be 3,1415927 nearly, which is the same as the rule.

II. As 22:7

or, as 355: 113

or, as 3.1415927: 1

}

:: the circumference: the diameter b.

EXAMPLES.-1. The diameter of a circle is 12, required the

Or, as 13.1415927 :: 12 : 31415927 × 12=37.6991124

the circumference very nearly.

2. The circumference is 30, required the diameter ?

30×7 105

Thus, as 22:7:: 30:

=

9.54545, &c. the dia

3. The diameter of a circle is 6, required the circumference? Ans. 18.8495562, &c.

4. The circumference is 5, required the diameter? Ans. 1.5915493, &c.

5. If the diameter be 100, what is the circumference? And if the circumference be 100, what is the diameter ?

288. To find the area of a circle.

RULE I. Let c=the circumference, d=the diameter, then

Or, 2nd. .7854 d2=the area. Or, 3rd. .07958 cthe area.

EXAMPLES.-1. The diameter of a circle is 4, required the circumference and area?

These proportions are the converse of the former,

Thus (Art. 252.) 3.1415927 × 4=12.5663708=the circum

Or, .7854 d2=.7854 × 16=12.5664 =the area, by rule 2. Or, .07958 c2=(.07958 × 12.5663709)2=.07958 × 157.913675, &c.=) 12.566769=the area, by rule 3.

2. Required the area of a circle, whose diameter is 7, and its circumference 22? Ans. 384.

3. What is the area of a circle, whose diameter is 1, and circumference 3.1415927?

289. To find the area of any irregular mixed figure ABCDEF. RULE I. Inscribe the greatest possible rectilineal figure ACEF in the proposed figure, and let ABC, CDE be the remaining irregularly curved boundaries.

quotient by the base AC, the product will be the area of the curved space ABC.

IV. Proceed in like manner, to find the area of the space CDE.

V. Find the area of the rectilineal figure ACEF by Art. 286. then lastly, add the three areas together, and the sum will be the area of the figure ABCDEF‹.

This method of approximation is used for measuring fields and other enclosures, which have very crooked and irregular boundaries; the greater the number of perpendiculars be, the nearer truth will the approximation be, as is evident.

To find the area of a regularly tapering board, measure across the two ends, add both measures together, and half the sum multiplied into the length of the board, will give the area.