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338. TO FIND THE AREA OF A CIRCLE.
1. Cut one half of a circular piece of paper as indicated in the diagram.
Observe that if the circle is cut into a very large number of parts and opened as shown in the figure, the circumference of the circle becomes, practically, a straight line.
Note. Imagine the circle cut into an infinite number of parts and thus opened and the circumference to be a straight line.
Observe that a circle may be regarded as made up of an infinite number of triangles whose united bases equal the circumference and whose altitude equals the radius. Hence to find the area of a circle we have the following:
RULE I. Multiply the circumference by 1 of the diameter.
2. It has already been stated that if the diameter of a circle is 1, its circumference is 3.141592. * Hence the area of a circle whose diameter is 1 is (3.141592 x ) .785398.
3. A circle whose diameter is 2, is 4 times as large as a circle whose diameter is 1 ; a circle whose diameter is 3, 9 times as large, etc.
Hence to find the area of a circle we have also the following:
RULE II. Multiply the square of the diameter by .785398.
4. The approximate area may be found by taking 1 (or .78) of the square of the diameter. See Werner Arithmetic, Book II., page 256.
* See page 229, note.
339. MISCELLANEOUS PROBLEMS. *
1. Find the approximate area of a circle whose diameter is 20 feet.
2. What is the area of a circle whose diameter is 1 foot ? 1 yard? 1 rod ? 1 mile?
3. What is the area of a circle whose diameter is 2 feet? 2 yards ? 2 rods ? 2 miles ?
4. A horse is so fastened with a rope halter that he can feed over a circle 40 feet in diameter. Does he feed over more or less than 5
rods ? 5. Find the approximate length (in rods) of the side of a square containing 2 acres.
6. Find the approximate diameter (in rods) of a circle whose circumference is one mile.
7. Find the approximate area of the circle described in problem 6.
8. Find the approximate circumference of a circle whose diameter is 30 rods.
9. The expression "a bicycle geared to 68” means that the machine is so geared that it will move forward at each revolution of the pedal shaft as far as a 68-inch wheel would move forward at one revolution. How far does a bicycle “geared to 68" move forward at each revolution of the pedal shaft? A bicycle "geared to 70" ?
10. What is the approximate circumference of the largest circle that can be drawn on the floor of a room 40 ft. by 40 ft. if at its nearest points the circumference is 2 feet from the edge of the floor?
*Require the pupil to make an estimate of the answer to each problem before attempting to solve it with the aid of a pencil.
340. The standard unit of volume measure is a cubic yard which is the equivalent of a 1-yard cube. This unit, like the cubic foot and the cubic inch, is derived from the corresponding unit of linear measure.
1728 cubic inches (cu. in.)
1 cubic foot (cu. ft.).
1. Show by a drawing that there are 27 cu. ft. in a 1-yard cube.
2. How many cubic inches in 1 half of a cubic foot ?
6. Estimate in cubic feet the amount of air in the school room.
7. Estimate in cubic yards the amount of air in the school room.
8. Estimate in cubic inches the capacity of your dinner box.
9. Estimate in cubic feet the capacity of some wagon box. 10. Estimate in cubic inches the volume of the school globe. *
* A globe is a little more than of the smallest cube from which it could have been made. See Werner Arithmetic, Book II., p. 266.
Denominate Numbers—Volume Measure.
341. Wood is usually measured by the cord. A cord is a pile 4 feet wide, 4 feet high, and 8 feet long or its equivalent. Hence
1. Estimate the number of cords of wood that could be put upon the floor of the school room if the desks were removed and the wood piled to the depth of four feet.
2. If 4-foot wood is piled 6 feet high what must be the length of the pile to contain 100 cords?
3. How many cords of wood in a pile 8 feet wide, 8 feet high, and 16 feet long?
4. Compare the amount of wood in the pile described in problem 3, with the amount in a pile one half as wide, one half as high, and one half as long.
5. If I pay $1.10 a cord for sawing wood, cutting each 4-foot stick into 3 pieces, how much ought I to pay for cutting each 4-foot stick into 4 pieces ?
6. A pile of wood 4 ft. high, 4 ft. wide, and 192 ft. long contains
cords. How many cords in a pile 4 feet high, 192 feet long, and 46 inches wide ?
7. A pile of wood is as wide as it is high and 32 feet long. It contains 9 cords. What is the width and height of the pile?
8. How many cords of 4-foot wood can be piled in a cellar that is 24 feet wide and 32 feet long, provided the pile is 4 feet high and one end of each 4-foot stick touches a wall of the cellar?
9. How many cords of wood in a pile that is 16 inches wide, 4 feet high, and 32 feet long ?
Denominate Numbers—Volume Measure.
342. Rough stone is usually measured by the cord. A pile 4 feet high, 4 feet wide, and 8 feet long or its equivalent, is 1 cord.
NOTE.-One cord of good stone is sufficient for about 100 cubic feet of wall. Hence in estimates it is customary to use the number 100 instead of 128; that is, as many cords of stone will be required for a given wall as 100 cubic feet is contained times in the number of cubic feet in the wall.
1. Estimate the number of cords of stone necessary for a cellar wall 18 inches thick, the inside dimensions of the cellar being 15 feet by 18 feet and 7 feet deep, no allowance being made for openings in the wall.
2. What are the outside dimensions of the wall of the cellar described in problem 1 ?
3. What length of wall 7 feet high and 18 inches thick is equivalent, so far as amount of stone is concerned, to the cellar wall described in problem 1?
4. If of the depth of the cellar described above is to be below the surface of the ground, how many cubic yards of earth must be excavated ?
5. How many per cent less of stone will be required for a 16-inch wall than for an 18-inch wall ?
6. Estimate the stone necessary for a wall 100 yards long, 11 feet high, and 2 feet thick.
7. If the specific gravity of stone is 2 and each cord is equivalent to 100 solid feet, how much does a cord of stone weigh?
8. If the specific gravity of a certain stone is 21, what is the weight of a block 8 feet by 2 feet by 2 feet ?