CONSTRUCTION. Draw the line AB, which make =106.5; and lay off 75.0 from A to m, at which point erect the perpendicular Cm 24.0; and join AC and B C, and you will have the triangle ABC.

With C as a centre, and the radius C n = : 25.8, describe an arc; and with A as a centre, and the radius An= 74.4, describe another arc cutting the former in n. Through the point n draw the diagonal AD = 108.0, upon which lay off Ao 36.0. At o erect the perpendicular Go= 22.5; join CD, DG, and G A, and the trapezium A CD G will be completed.

=

The trapezium DEFG may be constructed in a similar manner.

CALCULATION. Here 106.5 x 24=2556, double the area of the triangle ABC.

Again, 22.5+25.8 x 108 48.3 x 108-5216.4 double the area of the trapezium A CD G.

Also, 31.2+27.6 x 102-58.8 x 102-5997.6, double the area of the trapezium DE FG.

Then

2556+5216.4+5997.6__13770

2

2

•=6885 square in

ches, the area of the irregular polygon A BCDEFG. Lastly, 6885-282-24.414, the area in ale gallons; 6885-231-29.805, the area in wine gallons; and 6885÷ 2150.42=3.201, the area in malt bushels.

By the Sliding Rule.

In this operation, recourse must be had to the Rules given in Problems IV. and VI.

2. If the foregoing figure represent the base of a cooler; what quantity of ale does the vessel contain, when the depth of the liquor is 6.7 inches?

Ans. 163.5738 ale gallons.
L

3. Required the area of an irregular polygon in ale and wine gallons, and malt bushels; the first side measuring 94, the second 102, the third 64, the fourth 140, and the fifth 100 inches; and the diagonal from the first to the third angle 160, and that from the first to the fourth 130 inches.

Note. This figure is divided into three triangles; the areas of which may be found by Problem 5.

Ans. The area is 52.336 ale gallons, 63.890 wine gallons, and 6.863 malt bushels.

4. Required the area of the following irregular figure, in ale and wine gallons, and malt bushels; all the dimen sions being taken in inches.

Note 1. The dimensions of the foregoing figure, are taken as directed in Note 2; and evidently divide the figure into one right-angled triangle, and nine trapezoids. The base line is measured from A to B; the dimensions are entered from the bottom towards the top; the point on the base line, where each perpendicular rises to the opposite

angle, is entered in the middle column; and the perpendiculars themselves in the right and left-hand columns respectively. The base of each trapezoid may be found by subtracting the distances on the base line from each other; thus, the base of the first trapezoid on the left, is 260; the base of the first on the right, 335; the base of the second on the left, 460-260 200; the base of the second on the right, 555-335 = 220, &c. &c.

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2. When the dimensions are numerous, the foregoing method of entering them in the Note Book, will be found very convenient; if, however, a rough sketch of the figure be preferred, it may be drawn, and the dimensions entered upon the respective parts of the figure.

ANSWER.

Double Areas.

324337 trapezoids on the right.
656476 ditto on the left.

2)980813 sum.

490406 area in square inches.

Then, 490406282=1739.028, the area in ale gallons; 490406÷231=2122.969, the area in wine gallons; and 490406-2150.42=228.051, the area in malt bushels.

REMARK.

When any side of an irregular figure is curved, draw a chord-line so as to join the extremities of the curve; and erect perpendiculars from the chord-line to the curve, in such a manner as to divide the space contained between the chord and the curve into a number of small right-angled triangles and trapezoids, the areas of which must be found as in the last Example. Or the area of the space contained between the chord and the curve may be found by the method of Equi-distant Ordinates, described in Problem XX.

PROBLEM IX.

To find the area of a regular polygon.

RULE.

By the Pen.

Multiply the sum of the sides, or perimeter of the

Note 1. The area of a trapezium may also be found by dividing it into two triangles, and computing the area of each triangle by either of the last Problems.

2. Sometimes a trapezium may be very properly divided into two right-angled triangles and a trapezoid.

EXAMPLES.

1. What is the area of the trapezium A B CD, in ale and wine gallons, and malt bushels; the diagonal A C measuring 121 inches, and the perpendiculars BE, and DF, 38.3 and 43.1 inches respectively?