If the trapezium ABCD, (Fig. 33.) can be inscribed in a circle, the sum of the opposite angles BAD and BCD is 180° (Euc. 22. 3.) Therefore, the sine of BAD is equal to that of BCD or P/CD.

If s=the sine of either of these angles, radius being 1, and if AB-a, BC=b, CD-c. AD=d;

The triangle BAD-ads, And BCD=3bc×8; (Art. 9.) Therefore,

1. The area of ABCD=(ad+bc)xs.

To obtain the value of s, in terms of the sides of the trapezium, draw DP and DP' perpendicular to BA and BC. Then, Rad. s:: AD: DP :: CD: DP'.

Also, AP2=AD2-DP2, and CP/2-CD2-DP/2.

So that {DP-ADs=ds And

DP' CDX8=CS

But by the figure {

AP=vd2-d2 s2=dvl-s2

{

CP' vc2-c2 s2=c v1-s2

BP'=BC+CP'=b+c v 1—§3

BP-AB-AP-a-dv 1-s2

And BP2+DP2-DB2-BP/2+DP/2

That is a2-2 ad v 1—s2+d2=b2+2bc v 1—s2+c2

Substituting for s in the first rule, the value here found, we have the area of the trapezium, equal to

v (2 ad+2bc)3—(b2+c2—a2—d3)2

The expression under the radical sign is the difference of two squares, and may be resolved, as in Trig. 221, into the factors

(b+c3 ——a—d3)× (a+d2 —b—e3)

and these again into

(a+b+c¬d) (b+c+d—a)(a+b+d—c)(a+d+c—b) If then h-half the sum of the sides of the trapezium,

II. The area=√(h—a)×(h—b)×(h—c)×(h—d)

If one of the sides, as d, is supposed to be diminished, till it is reduced to nothing; the figure becomes a triangle, and the expression for the area is the same as in art. 10. See Hutton's Mensuration.

PROBLEM V.

To find the area of a REGULAR POLYGON.

15. MULTIPLY ONE OF ITS SIDES INTO HALF ITS PERPENDICULAR DISTANCE FROM THe center, AND THIS PRODUCT INTO THE NUMBER OF SIDES.

A regular polygon contains as many equal triangles as the figure has sides. Thus, the hexagon ABDFGH (Fig. 7.) contains six triangles, each equal to ABC. The area of one of them is equal to the product of the side AB, into half the perpendicular CP. (Art. 8.) The area of the whole, therefore, is equal to this product multiplied into the number of sides.

Ex. 1. What is the area of a regular octagon, in which the length of a side is 60, and the perpendicular from the center 72.126 ? Ans. 17382.

2. What is the area of a regular decagon whose sides are 46 each, and the perpendicular 70.7867?

16. If only the length and number of sides of a regular polygon be given, the perpendicular from the center may be easily found by trigonometry. The periphery of the circle in which the polygon is inscribed, is divided into as many equal parts as the polygon has sides. (Euc. 16. 4. Schol.) The arc, of which one of the sides is a chord, is therefore known; and of course, the angle at the center subtended by this arc.

Let AB (Fig. 7.) be one side of a regular polygon inscribed in the circle ABDG. The perpendicular CP bisects the line AB, and the angle ACB. (Euc. 3. 3.) Therefore, BCP is the same part of 360°, which BP is of the perimeter of the polygon. Then, in the right angled triangle BCP, if BP be radius, (Trig. 122.)

R: BP cot BCP CP. That is,

As Radius,

To half of one of the sides of the polygon;
So is the cotangent of the opposite angle,
To the perpendicular from the center.

Ex. 1. If the side of a regular hexagon (Fig. 7.) be 38 inches, what is the area?

The angle BCP='1⁄2 of 360°=30°. Then,

R: 19 cot 30° : 32.909-CP, the perpendicular,
And the area=19×32.909×6=3751.6

2. What is the area of a regular decagon whose sides are each 62 feet? Ans. 29576.

17. From the proportion in the preceding article, a table of perpendiculars and areas may be easily formed, for a series of polygons, of which each side is a unit. Putting R=1, (Trig. 100.) and n=the number of sides, the proportion be

And the area is equal to half the product of the perpendicular into the number of sides. (Art. 15.)

Thus, in the trigon, or equilateral triangle, the perpendic360° 6

ular-cot

=

cot 60°-0.2886752.

And the area=0.4330127.

In the tetragon, or square, the perpendicular

cot 45°-0.5. And the area=1.

In this manner, the following table is formed, in which the side of each polygon is supposed to be a unit.

By this table may be calculated the area of any other regular polygon, of the same number of sides with one of these. For the areas of similar polygons are as the squares of their homologous sides. (Euc. 20, 6.)

To find, then, the area of a regular polygon, multiply the square of one of its sides by the area of a similar polygon of which the side is a unit.

Ex. 1. What is the area of a regular decagon whose sides are each 102 rods? Ans. 80050.5 rods.

2. What is the area of a regular dodecagon whose sides are each 87 feet?

14

SECTION II.*

THE QUADRATURE OF THE CIRCLE AND ITS PARTS.

ART. 18. Definition I. A CIRCLE is a plane bounded by a line which is equally distant in all its parts from a point within called the center. The bounding line is called the circumference or periphery. An arc is any portion of the circumference. A semi-circle is half, and a quadrant one fourth, of a circle.

II. A Diameter of a circle is a straight line drawn through the center, and terminated both ways by the circumference. A Radius is a straight line extending from the center to the circumference. A Chord is a straight line which joins the two extremities of an arc.

III. A Circular Sector is a space contained between an arc and the two radii drawn from the extremities of the arc. It may be less than a semi-circle, as ACBO, (Fig. 9.) or greater, as ACBD.

IV. A Circular Segment is the space contained between an arc and its chord, as ABO or ABD. (Fig. 9.) The chord is sometimes called the base of the segment. The height of a segment is the perpendicular from the middle of the base to the arc, as PO. (Fig. 9.)

V. A Circular Zone is the space between two parallel chords, as AGHB. (Fig. 15.) It is called the middle zone, when the two chords are equal.

VI. A Circular Ring is the space between the peripheries of two concentric circles, as AA', BB'. (Fig. 13.)

VII. A Lune or Crescent is the space between two circular arcs which intersect each other, as ACBD. (Fig. 14.)

19. The Squaring of the Circle is a problem which has exercised the ingenuity of distinguished mathematicians for

Hutton's

*Wallis's Algebra, Legendre's Geometry, Book Iv, and Note IV. Mensuration, Horseley's Trigonometry, Book 1, Sec. 3; Introduction to Euler's Analysis of Infinites, London Phil. Trans. Vol. vi, No. 75, LXVI, p. 476, LXXXIV, p. 217, and Hutton's abridgment of do. Vol. 11, p. 547.