Ex. 3. Required the area of the trapezium ABCD, whofe diagonal AC is 20, and perpendiculars BE and DE, 8 and 10 Scots chains. Anf. 18 acres.

Ex. 4. In the trapezium ABCD, the fides AB is 45, BC 39, CD 42, DA 36, and the diagonal AC 48: Required the area. Anf. 1552, 7223.

Ex. 5. Required the area of the trapezium ABCD, whereof the fide AB is 10.25, BC 35, CD 50, DA 30, and the diagonal AC 40 chains. Anf. 76 acres 2 roods 19 falls 6 ells. Ex. 6. How many fquare yards paving are in a trapezium, whofe diagonal is 20, and perpendiculars 10 and 61 feet? Anf. 19 yards 2 feet.

Ex. 7. How many acres are in a field ABCD, of which the fide AB is 8000, AD 6000, and AC the diagonal 9560 links of the Scots chain: Alfo the angles* BAC, CAD are each of them 30°? Anf. 334 acres 2 roods 16 falls,

To find the area of a trapezium, its two diagonals and the included

It will be worth the learner's while to obferve, that when one of the angles of a right angled triangle is 30°, the leg oppofite to it will be exactly one half of the hypothenuse. Hence the perpendiculas BF and DF are 4000, 3000 the halves of the fides AB, AD.

EXAMPLE I.

Required the area of a trapezium, whose diagonals are 100, 80 feet, and the included angle 60°.

Ex. 2. Required the area of a, trapezium, whofe diagonals are 120, and 140 yards, and the included angle 30°.

Anf. 4200 fquare yards.

Ex. 3. What is the area of a trapezium, of which the diagonals are 80 and 60 Scots chains, and the included angle 60°? Anf. 207 acres 3 roads 8 falls.

If the trapezium be infcribed in a circle, its area may be found by the following rule.

Add all the four fides together; from half their fum fubtract the fides severally; then multiply the remainders continually into each other, and the square root of the last product will be the area.

EXAMPLE.

Required the area of a trapezium, whofe fides are 12, 13,

15 × 14 × 13X12-32760 and 32760=180,997 Anf.

PROBLEM

PROBLEM XI.

To find the area of an irregular polygon.

RULE.

Refolve the polygon into triangles by diagonals; find the area of each triangle separately, and their sum will be the area of the whole polygon.

EXAMPLE I.

Required the area of the following figure, ABCDEF, whose perpendiculars and diagonals are given. Fig. 78. plate 6.

Anf. 7 acres 3 roods 8 perches 243 yards.

The above example may be rendered more fimple, by redu

cing the figure to trapezias.

Anf. 7 ac. 3 ro 8 per. 24 yds. 7.80500 as before.

Required

Required the area of the irregular figure ABCDEF, of which the fide AB is 40, AC 5c, AD 55, AE 69, AF 36 Scots chains; and the angles are as follow. Plate 6. fig. 80.