SECTION VII. PRACTICAL MEASUREMENTS. MEASURES OF SURFACES. 139. A Surface is that which has length and breadth without thickness. THE RECTANGLE. 140. A Rectangle is a plane figure having four sides and four right angles; as, a slate, a door, etc. The Dimensions of a rectangle are its length and breadth. A rectangle has two bases; the side upon which it seems to stand is the lower base, and the side opposite the upper base. The Altitude of a rectangle is the perpendicular distance between its bases. A Square is a rectangle whose sides are all equal. The Perimeter of a rectangle is the sum of its sides. The Area of a rectangle is the number of square units it contains. B Thus, in the rectangle ABCD the area is the number of small squares it contains, which is equal to the number in each row multiplied by the number of rows, and this is equal to the number of linear units in the length multiplied by the number in the breadth. Hence the RULE. I. To find the area of a rectangle, multiply the length by the breadth. II. To find either side of a rectangle, divide the area by the other side. WRITTEN EXERCISES. 1. How many square feet in a yard 30 feet long and 20 feet wide? 2. How many square yards in a floor 33 feet long and 24 feet wide? 3. How many square rods in a field 120 rods long and 80 rods wide? 4. How many acres in a field 240 rods long and 120 rods wide? 5. How many acres in a field 15 chains long and 8 chains wide? 6. What is the width of a field 80 rods long which contains 12 acres? 7. How many square yards in a floor 16 ft. 6 in. long and 12 ft. 8 in. wide? 8. How many square yards in the sides of a room 24 ft. long, 20 ft. wide, and 12 ft. high? 9. A field is 10 chains long and 60 rods wide; how many acres does it contain? 10. How many chains long must a field 10 ft. wide be to contain an acre? 11. What will it cost to pave a sidewalk 75 ft. long and 6 ft. wide at 21 cts. a square foot? 12. What will be the cost of a concrete walk 120 ft. long and 4 ft. wide at 44 cts. a square yard? 13. What will it cost to cement the cellar of a house 40 ft. long and 24 ft. wide at 30 cts. a square yard? THE TRIANGLE. 141. A Triangle is a plane figure having three sides and three angles; as, CHK. The Base is the side upon which it seems to stand. The Altitude is the perpendicular distance from the vertex, K, to the base. The triangle M NO is seen to be one-half of the rectangle MNOP. If the rectangle is 8 units long and 8 units wide, it will contain 64 square units, and the triangle will contain 32 square units. This is equal to the number of linear units in the base multiplied by one-half the number in the altitude. Hence the M RULE. P Altitude K Base A Triangle. N I. To find the area of a triangle, multiply the base by one-half the altitude. II. To find the base or altitude of a triangle, divide the area by one-half the other dimension. WRITTEN EXERCISES. 1. What is the area of a triangle whose base is 35 inches and altitude 20 inches? 2. How many square yards in a triangular garden whose base is 45 ft. and whose altitude is 32 ft.? 3. How many acres in a triangular field whose base is 20 chains and altitude 15 chains? 4. A triangular field contains 16 A.; what is its altitude if it is 160 rods long? 5. What is the base of a triangular field whose altitude is 18 feet and whose area is 60 sq. yd.? 6. Find the area of a triangle whose base is 560 rods and altitude 90 chains. 7. Find the area in square feet of a triangle whose base is 16 feet and altitude 5 feet. 8. Find the area in square yards of a triangle whose base is 8 yards and altitude 6 feet. 9. What part of an acre is a triangle whose base is 82 feet and altitude 2 chains THE CIRCLE. 142. A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the centre. The Circumference is the bounding line; as, A D BE. Any part of the circumference, as BD, EF, etc., is an Arc. The Diameter is a straight line passing through the centre and terminating in the circumference; as, A B, DE. The Radius is a line drawn from the centre to the circumference; as, CD, C F, etc. E RULE. I. To find the circumference of a circle, multiply the diameter by 3.1416. II. To find the diameter of a circle, divide the circumference by 3.1416 or multiply the circumference by .3183. 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. To square a number, multiply it by itself. WRITTEN EXERCISES. Find the circumference of a circle whose 1. Diameter is 10 inches. 2. Radius is 10 inches. 3. Diameter is 31 feet. Find the diameter of a circle whose 5. Circumference is 10 feet. 6. Circumference is 40 inches. 7. Circumference is 80 rods. Find the area of a circle whose 9. Radius is 10 inches. 10. Diameter is 40 feet. 11. Circumference is 100 rd. 12. Radius is 18 feet. 13. The diameter of a circular fish-pond is 40 feet; what is its area? 14. Find the circumference of a circular window whose diameter is 3 feet. 15. A horse is tied to a stake by a rope 22 feet long; over how much space can he graze? 16. A pond 20 rods in diameter has a walk around it 6 feet wide; what is the area of the walk? 17 What is the area of a circle whose radius is a mile? 18. The area of a circle is 1 A. 154.16 P.; what is its circumference? 19. A wagon-wheel is 3 ft. 6 in. in diameter; how many miles does it travel in revolving 5000 times? |