1

EXAMPLES.

1. What is the area of a triangle whose base is 12 feet,

and altitude 3 yards?

3 yards=9 feet. Therefore of 12×9-54 square feet, or 6 square yards for the area.

2. What is the area of a triangle whose sides are re spectively 7, 11 and 12 feet?

SOLUTION.

of (7+11+12)=15 15x8x4x3=1440.

15-78

15-11-4

15-123

√1440=12√/10=37.95 nearly.

Hence the area is 37 95 sq. feet.

3. What is the area of a triangle whose base is 14 rods,

and whose altitude is 12 rods ?

Ans. 84 sq. rods.

4. What is the area of a triangle whose sides are respectively 13, 14 and 15 yards?

Ans. 84 sq. yards.

5. In a triangular field whose sides are 18, 80 and 82 feet,

how many square yards?

The area of any figure which is limited by any number of right lines, as the field ABCDEF, may be found by dividing it into triangles, and then computing each triangle separately, and taking their sum.

F

Ans. 80 sq. yards.

I'ROBLEM IV. To find the area of a trapezoid.

trapezoid into two triangles ABC, CDA. The area of the triangle ABC may be found (PROP. III.) by multiplying half the base A B into the altitude CF; and the area of the triangle CDA is found by multiplying half the base CD into the altitude AE, or into its equal CF. Hence the area of the trapezoid, which is the sum of the two-triangles, may be found by the following

RULE.

Multiply half the sum of the two parallel sides by the

half the sum of the par

allel sides, AD and BC, is found by measuring the width GH at the middle of the board. This average width, GH, being multiplied by the length EF will give its

area.

EXAMPLES.

1. If the parallel sides of a trapezoidal garden are re spectively 4 and 6 rods; and the perpendicular distance

between these sides is 8 rods, how many square rods in

the garden?

Ans.

{

40 sq. rods, or just of an acre.

2. How many square feet in a tapering board 16 feet long, measuring 15 inches wide at one end, and 10 inches at the other? Ans. 16 sq. feet.

PROBLEM V.-The diameter of a circle being given to find its circumference.

If the diameter of a circle is taken as a unit, the circumference will be 3.14159265, nearly. The exact value of the ratio of the circumference to the diameter has never been found. Its approximate value has been extended to more than 200 places of decimals. (Geometry, B. V, Prop. XIV, Scholium.)

Hence, when the diameter of a circle is known, its circumference may be found by the following

RULE.

Multiply the diameter by 3-1416.

333 355

22

NOTE. In the Higher Arithmetic, under Continued Fractions, we found some of the approximate values of this ratio to be 3, 27. ,, &c. This last value of 35 is true to six places of decimals. It may be easily retained in the memory by observing that if the first three odd numbers, 1, 3, 5, be duplicated, they will stand 113355. Now the first three figures give the denominator, and the other three give the numerator of the ratio.

EXAMPLE.-What is the circumference of the earth, on the supposition that it is 8000 miles in diameter?

Ans. 3-1416x8000=25132-8 miles, nearly.

PROBLEM VI.-To find the area of a circle, when its di ameter is known,

RULE.

Multiply the circumference by one fourth of the diameter. Or, what is equivalent, multiply the square of the diameter by of 3·1416. (Geometry, B. V., Prop. XI.)

0.7854

NOTE.-If a circle be inscribed in a square, its area will be to the area of the square, as 0-7854 is to 1.

EXAMPLES.

1. How many acres in a circle one mile in diameter? In a square mile there are 640 acres, therefore in a cir cle one mile in diameter there are

640 acres 0-7854-502-656 acres.

2. Which is the greater area, a circle 5 feet in diameter, or the sum of the areas of two other circles, the one being 4 feet in diameter and the other 3 feet?

whose diameter is A B, is found by multiplying its square by 0-7854. And the circle whose diameter is DE, is found by multiplying the square of this diameter by 0·7854. Hence, the difference of these areas is equal to the difference of the squares of the diameters multiplied by 0·7854.

PROBLEM VII.-To find the solidity of a prism, or of a cylinder.

RULE.

Multiply the area of the base by the altitude. (Geometry, B. VII., Prop. XI.)

EXAMPLES.

1. How many cubic feet in a rectangular stick of timber 10 inches by 12 inches, and 36 feet long?

=

10 inches of a foot, which is the fractional part of a square foot for the area of the end.

x36 30 number of cubic feet.

2. In a cylindrical log 14 feet long, and 14 inches in diameter, how many cubic feet?