SECTION VII.

PRACTICAL MEASUREMENTS.

466. The Applications of Measures to the farm, the household, the mechanic arts, etc., are so extensive that we now present a distinct treatment of the subject.

467. These Practical Measurements include Measures of Surface, Measures of Volume, and Measures of Capacity.

MEASURES OF SURFACE.

468. A Surface is that which has length and breadth without thickness.

469. The Area of a surface is expressed by the number of times it contains some other surface used as a unit of

measure.

THE RECTANGLE.

470. A Rectangle is a plane surface having four sides and four right angles. A slate, a door, the sides of a room, etc., are examples of rectangles.

471. A Rectangle has two dimensions, length and breadth. A Square is a rectangle in which the sides are all equal.

472. The Area of a rectangle is the surface included within its sides. It is expressed by the number of times it contains a small square as a unit of measure.

Rule I. To find the area of a square or rectangle, mul tiply its length by its breadth.

For, in the rectangle above, the whole number of little squares is equal to the number in each row multiplied by the number of rows, which is equal to the number of linear units in the length multiplied by the number in the breadth.

475. A triangle which has its three sides equal is called equilateral; when two sides are equal it is called isosceles ; when its sides are unequal it is called scalene.

Rule I. To find the area of a triangle, multiply the base by one-half of the altitude.

Rule II.-To find the base or altitude of a triangle, di vide the area by one-half of the other dimension.

EXAMPLES FOR PRACTICE.

1. What is the area of a triangle whose base is 15 ft. 6 in. and altitude 8 ft. 9 in.?

SOLUTION.-To find the area, we multiply the base by one-half the altitude; 15×43 = 6713 sq. ft., or 67 sq. ft. 117 sq. in.

2 How many square yards in a triangle whose base is 20 ft. 9 in., and altitude 10 ft. 11 in. ?

Ans. 12 sq. yd. 5 sq. ft. 371⁄2 sq. in. 3. Required the area of the gable end of a house 32.5 ft wide, the ridge being 15.25 feet above the wall.

Ans. 27 sq. yd. 4 sq. ft. 117 sq. in.

4. A triangular lot contains 233 sq. yd. 6 sq. ft. 108 sq. in.; its base is 165 ft.; what is its altitude? Ans. 25 ft. 6 in. 5. I have a triangular flower-bed containing 48 sq. ft. 63 sq. in., whose altitude is 6 ft. 3 in.; what is the base?

Ans. 15 ft. 6 in. 6. The gable end of a house contains 47 sq. yd. 6 sq. ft., the width of the house being 17 yd. 1 ft.; what is the height of the ridge?

THE CIRCLE.

Ans. 16 ft. 6 in.

476. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within, called Af

the centre.

C

An

477. The Circumference of a circle is the bounding line; any part of the circumference, as BC, is an Arc. are of one-fourth of the circumference is called a Quadrant. 478. The Diameter is a line passing through the centre

and terminating in the circumference; as, AB. The Radius is a line drawn from the centre to the circumference; as, OD. Rule I. To find the circumference of a circle, multiply the diameter by 3.1416.

Rule II. To find the diameter of a circle, multiply the circumference by .3183.

Rule III.-To find the area of a circle, multiply the circumference by one-fourth of the diameter, or multiply the square of the radius by 3.1416.

EXAMPLES FOR PRACTICE.

1. The diameter of a circle is 15 ft. 9 in.; what is its circumference?

SOLUTION. To find the circumference, we multiply the diameter by 3.1416; 3.1416×15-49.4802; hence the circumference equals 49.4802 ft.

2. What is the length of the tire of a carriage wheel 4 ft. 6 in. in diameter? Ans. 14.1372 ft. 3. In a square in a certain city is a fountain whose basin is 4 ch. 5 li. in circumference; what is its diameter?

Ans. 1 ch. 28.9115 li. 4. The end of the minute-hand of a church clock passes over 50 inches in 15 minutes; what is the length of the minute-hand? Ans. 31.83 in.

5. How much ground is occupied by a circular lighthouse, its circumference being 50 ft.? Ans. 198.9375 sq. ft. 6. Within a circular plot 50 rods in diameter is a circular pond, whose edge is everywhere 6 rods from the edge of the plot; what is the area of the pond? Ans. 1134.1176 P. 7. A walk 3 ft. wide extends around the above mentioned plot; what is the area of the walk? Ans. 7803.7344 ft.

MEASUREMENT OF LAND.

479. The Unit of Measure of land is the Acre, which is sometimes divided into square rods and sometimes into square chains. Hundredths of an acre are also frequently used

In 1802, Col. Jared Mansfield, Surveyor-General of the North-Western