Rule. Find the area of one of the triangles (by prob. 4), and multiply that area by the number of triangles : Ör x half the sum of the fides by a line drawn from the middle of any one of the fides to the center of the polygon, and the product will be the area.

Note. The radius of a circle infcribed in any regular po• lygon is equal to the perpendicular of it, let fall from the center to the middle of the fide of the polygon. Now by plane Trigonometry a Table may be calculated for the more ready meaiuring of regular polygons; which Table I fhall here fub join; expreffing the angles at the center, length of the perpendicular (orradius of the circle inscribed in it) ard area.

The names Number The angle] The perpen-[The area. The

dicular. fide of the Polygon being 1.

The use of the preceding Table.

It is well known to Geometricians, that the areas of fimi. lar (or like) plane figures are in proportion to one another, as the fquare of their correfponding fides: Therefore multiply the fquare of the fide of a given regular Polygon by fuch a number taken out of the above table, as is agreeable to the name of the Polygon, and the product will be the area thereof, in the fame denomination as the given fide.

Ex. If the fide of a regular Octagon be 12 feet; what is its area?

Operation 4.8284271×12×12=695.2935024 the area, 7. To Meafure a Circle.

g

Definition. A Circle is a plane figure, bounded by one continued line, called the circumference, or periphery; every part of which is equally di stant from a point within the Circle, called its Center; from which, any right line (ac, cd, &c.) drawn to the cir. cumference, is called the radius, or semi-diameter of the Circle ; any right line ab, drawn thro' the center, terminating each way at the circumference, is

b

called a diameter; a right-line dg, lefs than the diame. ter, meeting the circumference, in two points, is called a chord, or fubtenfe; and the perpendicular distance rp from the middle of the chord to the circumference is called a verfed fine.

Before I proceed to find the area of a 'Circle, it will be neceffary to fhew the learner, how to find the circumference of a Circle, by having its diameter given, and the contrary.

It is now looked upon, even by Mathematicians of the first rank, as abfolutely impoffible to determine the exact proportion of the diameter and circumference of a Circle.

That great Geometer Archimedes, about 2000 years ago, first discovered this proportion to be nearly as 7 to 225 that is, if the diameter of a Circle be 7, its circumference will be 22, very nearly. But 22 is too much.

Metius Snellius difcovered the proportion to be nearly as 113 to 335; which is nearer the truth than Archimedes, but 335 is too great.

Since Archimedes's and Metius's time, various me. thods have been invented, whereby the faid proportion may be approximated to a very great degree of exactnefs.

Van Ceulen (a Dutchman) found by incredible pains, that if the diameter of a Circle be represented by i, the circumference thereof will be

3,14159265358979323846264338327950288,

extreamly near. This laft number was not only confirmed, but was extended to double the number of decimal places, by that ingenious and moft indefatigable Mathematician, the late Mr. Abraham Sharp, of Little Horton, near Bradford, in Yorkshire. And Mr. John Machin, Professor of Astronomy in Gresham College, and Secretary to the Royal Society, has carried it to 100 places. But in the ordinary practice of meafuring, it will be unneceffary to take any more than 3,14159 (or 3,1416): Hence it is evident, that if the diameter of any circle be multiplied by 3,1416, the product will be the circumference of that circle, very nearly.

Ex. What is the circumference of a circle, whofe diameter is 50?

Operation. 3,1416×50-157,08 the circumference.

It is evident, from this example that if the circumference of a circle be divided by 3,1416, the quotient will be the diameter. For 157,08 3,14,650 the diameter. To find the area of a circle, by having the diameter and circumference given.

Rule. Multiply half the circumference by half the

C c

dia

diameter, and the product will be the area. For every circle is equal to a rectangled parallelogram contained un der half the circumference and half the diameter.

Ex. What is the area of a circle whofe diameter is 1, and circumference 3,1416?

Operation. the circumference is 1,57c8, and the diameter is .5 Then 1,5708 X,5,7854 the area of a circle whofe diameter is 1, nearly.

Therefore if the fquare of the diameter of any circle te multiplied by 7854, the product will be the area, or measure of the circle in that denomination whereby the diameter was expreffed, whether inches, feet, yards, &c.

As for inftance, fuppofe the diameter of a circle be 50 feet, the fquare whereof is 2500; then 7854 X2500 1963,5 the area fought, nearly.

The circumference of a circle being given to find the area. Rule. Multiply the fquare of the circumference by 0,0795776 (1,3,1416x4) and the product will be

the area.

Ex. If the circumference he 39, what is the area? Operation. 0,0795776×90071,61984 the area. The area of a circle being given to find the diameter. Rule. Multiply the square root of the area by 1,12837 (=21/1÷3,1416) and the product is the diameter. Ex. What is the diameter of a circle whose area is 2500?

Operation 1,12837 X250056,4185 the diameter. The diameter of a circle being given to find the fide of the greatest infcribed Square.

Rule. Multiply the given diameter by,707 (=√1), and the product will be the fide of the fquare required. Ex. If the diameter of a circle be 50; what is the fide of the infcribed fquare? Anfw. 35,35

8. To measure the sector of a Circle.

Definition. A Sector of a Circle drge, is a figure con.

tained by an arch (or arc) thereof, and two radius's i when these two radius's form a right angle, or the arch becomes 4th of the circumference, the figure is called a quadrant, as adre, (or ergb, fee the laft figure)..

Rule. Multiply half the length of the arch, by the radius, and the product will be the area.

Ex. Let drge reprefent a fector (or quadrant) of a circle, whofe radius de (or ge) is 45 feet, and length of the arch drg, 70,686 feet, required its area.

Operation. 45×70,586—1590,435, the area.

By this propofition the area cf the fegment of a circle may be found: For if the area of the triangle dgc ve fubtracted from that of the fector drge, there will remain the area of the fegment drgd. As for instance, fuppofe the chord og 63,5; feet, and the perpendicular c= 3,8 5. Then 63,63X31,815÷2=1012.194225 the area of the triangle deg, which taken from 1599,435 (the area of the lector) eaves 578,240775 the area of the fegment (drgd). Or the area of the fegment of a circle may be found by having the chord and versed fine given, by the following general rule.

Rule. Multiply the verfed fine (or height) by ,626; to the fquare of the product add the fquare of half the chord: multiply twice the fquare root of the fum, by two thirds of the verfed fine, and the product will be the area.

Ex. Required the area of the fegment of a circle, whofe chord is 48 and verfed fine 18.

Opera, 18x,626=11,268, whofe fquare is 126,967824 which added to the fquare of half the chord (576), makes the fum 702,96782; twice the fquare root of which, is 53,026; which xe by two thirds of the height (12), the product is 636,312 the area.

If the fum of the fquares of the femi-chord and verfed fine be divided by the verfed fine, the quote will be the diameter of the circle, to which that fegment correfponds. As for inftance, suppose the chord be 24, and the versed fine 8. Then 12×12+8×8÷÷8=26 the diameter required. Cc a

By