15. Construct a square. Construct its radius and its apothem.

16. Inscribe a regular hexagon in a circle. Construct its radius and its apothem.

17. A number 67 hat is 67 in. in diameter. Find the length of the hat band.

18. The steamship Carmania sailing from Liverpool to New York was at noon on Aug. 27 in longitude 9° 19′ W. .; Aug. 28, 20° 35′ W.; Aug. 29, 30° 52′ W.; Aug. 30, 39° 47′ W.; Aug. 31, 47° 55′ W.; Sept. 1, 57° 06′ W.; Sept. 2, 66° 09′ W. Find the number of degrees of longitude through which the ship sailed each day.

19. Do you know of any church windows that are decorated with designs which resemble Figure 94?

20. Measure the radii of the arcs in Figure 93. Find the sum of the lengths of all the arcs.

CHAPTER XI

AREAS

98. Comparison of rectangles.

1. Make a rectangle R with its length equal to a given line 7 and its width equal to a given line w. Make another rectangle, R', with its width w and its length 37. How many rectangles the size of R can be cut from rectangle R'?

2. Make another rectangle with length and width 5w. How many rectangles like R can be cut from this rectangle?

3. Construct a rectangle with length 27 and width 3w. This new rectangle is how many times the rectangle R? Show by cutting the new rectangle.

4. A rectangle R has a length b and a width a, and rectangle R' has a length 4b and width 3 a. Rectangle R' is how many times rectangle R?

5. Draw a line a, 1 inch long. Construct a rectangle, R, with length 3a and width 2 a, and a rectangle, R', with lengtha and width a. Show the number of times rectangle R contains rectangle R'.

6. Show by a drawing how many times a 2-inch square is contained in a 4-inch square; in a 6-inch square; in a 10-inch square.

7. What part of a square inch is a rectangle whose sides are in. and in.?

8. A rectangle 8 units long and 31⁄2 units wide contains the square whose side is 1 unit how many times?

9. If the length and width of a rectangle are measured by any unit of length, how can you find out how many times the rectangle contains a square whose side is 1 unit long?

45

10. The length of a rectangle is ft. and its width is ft. Its area is what part of a square foot?

B

C

FIG. 95.

11. Make a rectangle whose length is 3 in., and its width 23 in. Show by a drawing that its area

is 23×3 sq. in.

Exercise 91

A yard of carpet means a strip one yard long but of any width which it happens to be made.

Wall paper is bought in rolls containing either 8 yards or 16 yards. It is usually 18 in. wide but is sometimes made other widths.

1. A certain room is 15 ft. long and 12 ft. wide. It is to be covered with carpet in strips 3 ft. wide, cut as long as necessary. If the strips are laid lengthwise of the room, how long must each strip be? How many strips will be needed? If laid crosswise of the room, how many strips How long must each strip be?

will be needed?

2. A floor is 17 ft. long and 15 ft. wide. How many strips each a yard wide will be required to cover the floor if laid crosswise? If laid lengthwise? If the strips are laid crosswise what part of a strip must be cut off or turned under?

3. A floor is 16'X20'. Draw it to the scale of 1 ft. to in. Show the number of strips of matting 3 ft. wide which are required to cover it, if laid lengthwise; if laid crosswise. In which case would there be the more waste?

Compute the number of yards of carpet required for the following rooms when the strips are laid lengthwise, also when laid crosswise, and decide which way requires the smaller amount of carpet. If in doubt, make drawings showing the carpet laid each way.

4. A room 10'x15', carpet a yard wide. 5. A room 13'x18', carpet a yard wide. 6. A room 7' 6"x10', carpet 2' 6" wide.

7. A room 16'x18' 6", carpet a yard wide. 8. A hall 8'x22' 4", carpet 2' 9" wide.

9. A kitchen 11'X14' 6", with linoleum 6' wide.

A roll A roll

10. A wall 7' 6" high and 16′ long is to be papered. of the paper used is 18" wide and contains 8 yd. makes how many strips? Pieces of strips cannot be used. How many rolls are required?

11. The hearth before a fireplace is 6' long and 2′ wide. It is laid with bricks each 8"x4". If the bricks are laid lengthwise of the hearth, how many bricks are required along the long edge of the hearth? How many rows of bricks are required to cover the hearth? How many bricks are required to cover the hearth?

If laid crosswise of the hearth, how many bricks in each row? How many rows? How many bricks to cover the hearth?

12. A hearth 5'x1'43" is to be covered with tiles each 6"X12". How many tiles will be required to cover it?

13. If shingles have an average width of 4 in., how many are required for one row on a roof 22 ft. long?

Shingles are laid in rows overlapping each other. If three inches of the length of a shingle are not covered by the overlapping shingle, they are said to be laid 3 inches to the weather.

14. If the shingles are laid 4 in. to the weather, how many rows of shingles must be laid to cover a roof 12'X22'? How many shingles? There is a double row at the bottom.

15. How many shingles are required for a roof 10' 6"×28′, laid 5" to the weather, the average width of a shingle being 4"?

16. A certain roof requires 3560 shingles if they are laid 4 in. to the weather. How many does it require if they are laid 3 in. to the weather?

17. A field is 40 rd. X80 rd. It is planted in corn in rows 3′ 6′′ apart. The hills of corn are 3' apart in the rows. How many hills of corn are there in the field?

18. If tomatoes are planted in rows 5' apart with the plants 4' apart in the rows, how many plants are there in a field 30 rd.X48 rd.?

19. How many paving bricks each 8"x4"X3" will be required to pave a street in front of a lot 40' wide, if the street is 30' wide and the bricks are laid with the face 3"X8" up? If laid with the face 4"X3" up?

99. Area of a rectangle. If the length and width of a rectangle are measured by the same unit of length, it is convenient to measure its area by a square whose side is that unit of length.

The pupil has seen that the number of unit squares is the product of the number of linear units in the base by the number of linear units in the altitude. This is what is meant by the formula

A = ab, and by the

Rule. To find the area of a rectangle, multiply its base by its altitude.