Note. In measuring boards, you can multiply the length ir feet by the breadth in inches, and divide by 12; the quotien will give the answer in square feet.

5. In a board 20 feet in length, 16 inches in width, how many feet? Ans. 20×16=320÷12=263 feet.

To measure a triangle, or to find the area

A triangle is a figure bounded by three straight lines; thus, B, A, C, is a triangle. When a line like A, D, is drawn, making the angle A D B square to the angle A D C, then A D is said to be perpendicular to B C, and A Dis called the altitude of the triangle. Each triangle, BAD, or D A C, is called a right-angled triangle. The side B A, or the side A C, opposite the right angle, is called the hypotenuse.

RULE.

C

hyp.

A

per.

base

line.

D

B

Multiply the base of the given triangle into half its perpendicular height, or half the base into the whole perpendicular, and the product is the answer.

1. Required the area of a triangle whose base or longest side is 32 inches, and the perpendicular height 14 inches.

Ans. 32×7 (of 14)=224 square inches. 2. In a triangular field the base is 40 chains, and the perpendicular 15 chains; how many acres?

=

Ans. 15 x 40-600-2=300÷10 ch. 30 acres. 3. What is the area of a square piece of land, of which the sides are 27 chains? Ans. 72 acres, 3 roods, 24 poles. 4. How many acres in a piece of land 560 rods long and 32 rods wide? Ans. 112 acres. 5. How many acres are contained in a road 40 miles long and 4 rods wide? Ans. 320. 6. What will a lot of land 1 mile square come to, at D20.75 per acre? Ans. D13280.

7. How many yards of carpeting, that is 14 yards wide, will Bover a floor 21 feet, 3 inches long, and 13 feet, 6 inches wide? Ans. 25 yards.

To measure a circle, area, circumference, &c.

E

C diameter

radius

H

B

A circle is a portion of a plane bounded by a curved line, every part of which is equally distant from a certain point within, called the céntre. The curved line, A E B D, is called the circumference; the point A C the centre; the line A B, passing through the centre, a diameter, and C H the radius. The circumference A E B D is 3.1416+times greater than the diameter A B. Hence, if the diameter is 1, the circumference will be 3.1416+ Also, if the diameter is known, the circumference is found by multiplying 3.1416 by the diameter. Hence the following rule:

RULE.

circumference

Multiply the diameter by 3.1416, and the product will be the circumference; or divide the circumference by 3.1416, and the quotient will be the diameter.

Note 1.—As 7 is to 22, so is the diameter to the circumference; or, as 22 is to 7, so is the circumference to the diameter. Or, more correctly:

As 113 is to 355, so is the diameter to the circumference; or as 355 is to 113, so is the circumference to the diameter.

Note 2.-The problem of "squaring the circle," as it is usually termed, has never been solved, nor can a square or any other right-lined figure, be found, that shall be equal to a given circle, as there must be a fraction in one case or the other; it is not in the power of numbers to bring them exactly alike; the numbers used above are sufficiently correct; but the calculation may be extended to almost an indefinite number of decimals, with-out apparently arriving any nearer the solution of the problem. 1. If the circumference of a circle be 354, what is the diameter? Thus 354.000÷3 1416=112.681, diameter. Ans.

2. If the diameter of a circle be 17, what is the circumference? Thus: 3.1416×17=53.4072, circumference. Ans. 3. If the circumference of the earth be 25000 miles, what is the diameter ? Ans. 7958 miles (nearly). 4. The base of a cone is a circle; what is its diameter, when the circumference is 64 feet? Ans. 20.3718

5. What is the circumference of a wheel whose diameter is 5 feet, 2 inches? Ans. 16 2316.

6. If the circumference of a carriage-wheel be 16 feet, 6 inches, what is the diameter ? Ans. 5.2521 feet,

7. The circumference of a circle is 16 chains; what is the diameter ? Ans. 5.0929+chains.

The diameter given, to find the area or content.

RULE.

Multiply the square of the diameter by the decimal .7854, and the product will be the area.

1. How many square feet are contained in a circle, whose diameter is 4 feet 3 inches? Thus 4.252=180625 ×.7854=

14.1862875 square feet. Ans.

2. What is the value of a circular garden, whose diameter is 6 rods, at the rate of 8 cents per square foot ?

Ans. D615.81.6.432.

The area af a circle given, to find the diameter.

RULE.

Divide the area by .7854, and extract the square root of the quotient.

1. The area of a circle is 5 acres, 3 roods, 26 perches, required the diameter. Thus: 946.000.7854-34.7 poles. Ans. 2. What is the length of a rope fastened to a stake in the centre of a circular field, and the other end to the nose of a horse, which will permit him to feed on 2 acres of land?

Ans. 2.5231 chains.

The circumference given, to find the area.

RULE.

Multiply the square of the circumference by the decimal .07958, and the product will be the area.

If the circumference of a circle be 1, the diameter 1÷ 3.14159=0.31831; and the product of this into the circumference is .07958, the area.

1. If the circumference of a circle be 136 feet, what is the area? Ans. 1472 feet. 2. The circumference of a circle being 37.7, required the area. Thus: 37.72=1421.29=square, .07958 113.1062582 area of the circle.

To find the surface of a sphere, globe, or ball.

Square the diameter, and multiply it by the decimal 3.1416, and the product will be the answer.

1. What is the surface of a sphere whose diameter is 12? Ans. 122-144 x 3.1416-452,3904

2. Required the number of square inches in the surface of a sphere whose diameter is 2 feet, or 24 inches? Ans. 1809.5616.

To find the solidity of a sphere.

A sphere or globe is a round solid body.

RULE.

Sphere.

Multiply the surface by the diameter and divide the product by 6; the quotient will be the solidity (multiply the square of the diameter by 3.1416).

1. What is the solidity of a sphere whose diameter is 12? Thus: 122-144×3.1416=452.3904 × 12=5428.6848÷6= 904.7808, Ans. Or, multiply the cube of the diameter by .5236, thus: 12 x 12 x 12 x .5236-904.7808.

2. What is the solidity of the earth, its mean diameter being 7918.7 miles? Ans. 259992792082.6374908.

To find the solidity of a prism. Definition.—A prism is a body with two equal or parallel ends, either square, triangular, or polygonal, and three or more sides, which meet in

Prism.

parallel lines, running from the several angles at one end to those of the other.

RULE.

Multiply the area of the base by the altitude, and the product will be the content.

1. What is the content of a prism, each side of the square which forms the base being 15, and the altitude of the prism 20. feet? Ans. 152-225 ×20=4500 feet. 2. The side of a stick of timber is hewn 3 square, is 10 inches, and the length is 12 feet; required the content.

Ans. 10×4.33= perp.=43.3 area at the end, ×12 feet length, 519.6÷144=3.6+ content.

3. Required the solidity of a triangular prism whose height is 10 feet, and area of the base 350 feet. Ans. 3500 feet

To find the convex surface of a cylinder. Definition. A cylinder is a round body, whose bases are eircles, like a round column, or stick of timber, of equal bigness at both ends.

RULE.

Cylinder.

Multiply the circumference of its base by the altitude.

1. What is the convex surface of a cylinder, the diameter of whose base is 20, and the altitude 50 feet?

Ans. 3.1416 x 20 × 503141.6000

2. Required the convex surface of a cylinder, the circumfer ane of whose base is 6509, and altitude 27.

Ans. 175743

To find the solidity of a cylinder.

RULE.

Multiply the square of the diameter of the end by .7854, which will give the area of the base; then multiply the area of the base by the length, and the product will be the content.

1. Required the solid content of a round stick of timber of equal bigness at both ends, whose diameter is 1.5 and length 20 feet. Thus 1.5 x 1.5=2.25 × .7854 ×20=35.3430. Ans. Or, 18×18=324 × .7854 × 20=5089.3920÷144-35.3430. Ans.

2. What is the solidity of a cylinder, the diameter of whose base is 12, and the altitude 30? Ans. 3392.928. 3. How many solid feet in a round stick of timber 16 feet long, and the diameter at each end 15 inches?

Ans. 19.635 solid feet. 4. Required the solidity of a cylinder, the diameter of whose base is 30 inches, and the height 50 inches?

Ans. 20.4531 solid feet

To find the solidity of a cone.

Definition. A cone is a round solid body of a true taper from the base to a point, which is called the vertex.

Multiply the area of the base by the altitude, and divide the product by 3; that is, square the diameter and multiply it by .7854, which gives the area of the base; then multiply by the altitude and by 3. Or, the square of the circumference of the base by .07958, and that product by of the perpen dicular altitude, and the product will be the solidity.

1. Required the solidity of a cone, the diameter of whose base is 5 and the altitude 10.

Ans, 52-25 x .7854-19.635 × 10÷3=65.45 2. What is the solidity of a cone whose altitude is 27 feet and the diameter of the base 10 feet?

3. Required the solidity of a cone, the base is 18 inches, and its altitude 15 feet.

Ans. 706.86. diameter of whose

Ans. 8.83575 feet.

4. If the circumference of the base of a cone be 40 feet, and the height 50 feet, what is the solidity? Ans. 2122.1333 feet 5. The top of a cistern is 5.5 feet, the bottom 4.75 feet, and the height, or depth, 7.25; how many hogsheads will it con Ans. 17.75

tain?