Section 17.

40. Find the area of a triangle, in acres, whose sides are 981, 84}, and 864 yards.

41. Find the area of a triangle, in acres, whose sides are 108, 144, and 216 yards.

42. Find the area of a triangle, in acres, whose sides are 20, 30, and 40 feet.

Section 18.

43. The sides of an equilateral triangular field are each 8 chains what is its value at 84l. per acre?

;

44. A triangular plot of ground measures along its three sides, 24 yards 2 feet, 28 yards 1 foot, and 34 yards; what is its value at 26s. per square yard?

PROBLEM 2.

To find the area of a trapezium, and also of any rectilineal figure.

Divide it into two triangles by either of the diagonals A C or B D, then find the area of each of the triangles by Prob. 1.

A

Any rectilineal figure of any number of sides can be divided into triangles whose areas may be found by Prob. 1.

Then, 13 14 × 4.7727 = 62 713 square chains, = 6.2713 acres = 6 acres, 1 rood, 3 perches, 124 square yards.

EXAMPLES.
Section 1.

1. The two sides of a rectangle are 14 feet 6 inches, and 3 feet 5 inches; how many square feet does it contain ?

2. A garden whose form is a parallelogram, one side of which is 3 chains, and the perpendicular between the sides 2 chains; how many roods does it contain?

3. A balk of timber is 36 feet 8 inches long, and 2 feet 6 inches wide; what is the value of it at 8d. per square foot ?

Section 2.

4. An oblong room is 36 feet 8

inches long, and 18 feet 7 inches wide; how many yards of carpet, 2 feet 6 inches wide, will cover it? and what is the value of the carpet at 5s. 10d. per yard?

5. A square field containing 5 acres is to be fenced round at the rate of 2s. 6d. per lineal yard; what is the price of fencing it?

Section 3.

6. An oblong field is 8 chains 27 links long, and 5 chains 34 links wide; what is the value of it at 100l. per acre?

7. How many square feet of timber will make a chest 6 feet 6 inches long, 2 feet 3 inches wide, and 2 feet 6 inches deep? 8. A parallelogram is 14 feet 6 inches long, and 5 feet 3 inches between the sides; find its area in square feet.

Section 4.

9. A rectangular field is 4 chains 73 links long, and 5 chains 83 links wide; how many acres does it contain ?

10. A rectangular field is 15 chains and 63 links long, and 12 chains 53 links wide; what is the rental of it at 31. 15s. per acre?

11. A rectangular field is 8 chains 73 links long, and 10 chains 43 links wide; what will be the price of mowing it at 9s. 6d. per acre?

Section 5.

12. Find the square feet in the following four rectangular pieces of ground: 14 feet 6 inches long, and 5 feet 3 inches wide; 10 feet 6 inches long, and 4 feet 4 inches wide; 14 feet 6 inches long, and 3 feet 5 inches wide; and 15 feet 6 inches long, and 3 feet 4 inches wide.

Section 6.

13. Find the contents of each of the three rectangular fields: 7.34 chains long, and 3.54 broad; 3.14 chains long, and 2.24 broad; and 5.16 chains long, and 3.89 broad.

Section 7.

To find the area of AD B C; add the heights AD and B C, and divide by 2 for a mean height; multiply the mean height by the length A B, and the result will be the area.

D

A

B

14. How many acres are there in a field, whose length is 13.14 chains, and perpendicular breadths at each extremity 132 yards, and 78 yards respectively.

15. Find the value of a piece of timber, whose dimensions are 14 feet 7 inches long, 3 feet 4 inches at one end, and 1 foot 4 inches at the other, at 2s. 7d. per square foot.

16. Find the value of a field, whose length is 33.74 chains, and perpendicular breadths at each extremity are 3.42 chains, and 2.23 chains respectively, at 571. 17s. 6d. per acre.

Section 8.

17. The area of a field is 1 acre, whose length is 11

chains,

and breadth at one end chain; what is the breadth at the other end?

18. A piece of timber contains 30 square feet; find the length, the perpendiculars at each end measuring 14 inches and 18 inches respectively.

19. A garden, which contains a rood, has its length equal to its greatest perpendicular length at one extremity, and the perpendicular at the other extremity is half the former; find the length.

PROBLEM 4.

To find the area of a circle, ellipse, and parabola.

Circle.

Rule I.-Square the diameter and multiply the result by .7854 for the area.

Rule II.-Square one fourth of the circumference, and divide the result by .7854 for the area.

Rule III.-Take half the circumference and half the diameter, and multiply them together for the area.

Rule IV.-Square the radius, and multiply the result by 3.1416 for the area.

Ellipse.

Rule.-Multiply the diameters and .7854 together for the area.

Parabola.

Rule.-Multiply the abscissa by the double ordinate; of this product will be the area.

FORMULE.

Let D diameter of circle, c = its circumference, and r = its radius.

a, b the diameters of the ellipse.

x, y

abscissa and double ordinate of parabola.

It is readily seen that a circle is a polygon of an infinite number of sides, the circumference being the perimeter, and the radius the perpendicular. Hence, the area of a circle is the area of a triangle whose base is the circumference, and perpendicular height is the radius.

The squaring of a circle is a very ancient problem, the object of which is to find a square, geometrically, equal in area to a circle.

This problem, in consequence of its great difficulty, has engaged the attention of thousands in every age since the Greek epoch without the slightest success; but the labours of Van Ceulen, Sharp, and Rutherford have approximated to the solution so near, that the exact solution would not be attended with any practical benefit.

1. Find the area of a circle, whose diameter is 15 inches.

2. Find the area of a circle, whose diameter is 8 inches. 3. Find the area of a circle, whose diameter is 6 feet 7 inches. 4. Find the area and circumference of a circle, whose diameter is 15 inches.

Section 2.

5. Find the area of a circle, whose circumference is 29.8452 inches.

6. What is the diameter and area of a circle, whose circumference is 251.328 inches?

7. What is the area and circumference of a circle, whose diameter is 7 feet 4 inches?

Section 3.

8. Find the areas and circumferences of the circles, whose diameters are 8 feet, 9 feet, 15 feet, 6 feet, and 24 feet.

Section 4.

9. Find the areas and circumferences of the diameters are 12 feet, 134 feet, 16 feet, 21

feet.

circles, whose

feet, and 345