The product of the number of linear units in the two dimensions of a parallelogram is the number of square units in its area.

That is,

A = 1X w.

NOTE. When letters are used, the sign of multiplication may be omitted. Thus A = 1 × w may be written A = lw.

5. Find the area of a parallelogram whose base is 24 ft. and whose altitude is 15 ft.

6. A road 3 rd. wide runs diagonally across a farm. The length of the road is 36 rd. What part of an acre in the road?

7. A man has a plot of land between two parallel roads that is in the form of a parallelogram with a base of 36 rd. and a width (altitude) of 24 rd. At $120 per acre, what is the plot worth?

8. Ralph has a garden plot in the form of a parallelogram. One pair of parallel sides is 80 ft. long and the distance between them is 45 ft. How many square feet in the garden?

FINDING THE AREA OF A TRIANGLE

A figure inclosed by three straight lines, as in the figure, is called a triangle.

1. Draw upon the blackboard triangles

of different shape.

2. From a piece of tablet paper or cardboard cut two triangles exactly alike.

3. Place two equal sides together so as to form a figure as shown on the next page. What is the figure called?

FINDING THE AREA OF A TRAPEZOID

299

4. If the base and altitude of the triangles are 10 in. and 6 in. respectively, what are the di

mensions of the parallelogram they form?

5. What is the area of the par

allelogram? Of one of the two triangles that form the parallelogram?

From this we see that

One half the product of the number of linear units in the two dimensions of a triangle is the number of square units in its area.

Calling A the number of square units in the area, b the number of linear units in the base, and h the number of linear units in the altitude, this is expressed by the formula

6. How many square inches in the area of a triangle whose base is 12 in. and whose altitude is 9 in.?

7. A creek cut off a triangular plot from Mr. Barnes's farm. This plot which formed a triangle with a base of 8 rd. and an altitude of 12 rd. was used as a truck patch. What part of an acre in the plot?

FINDING THE AREA OF A TRAPEZOID

A figure like the one shown here is a trapezoid. It has two parallel sides called its bases. The distance between the bases is its altitude. To measure it we

have to find its relation to a known parallelogram.

1. Cut from paper or cardboard two

trapezoids exactly alike. Place the two as in the figure on the next page.

2. What is the shape of the figure formed by the two trapezoids? What is its base? Its altitude?

3. If the upper base of the trapezoid is 5 in., the lower base 7 in., and the altitude 4 in., what are the dimensions of the parallelogram formed by the two trapezoids? What is the area?

4. What, then, is the area of one of the trapezoids?

5. Then tell how to find the area of a trapezoid.

One half of the product of the number of linear units in the altitude by the sum of the number of linear units in the two bases of a trapezoid is the number of square units in its area.

Calling the upper base b1 ("b sub one"), the lower base b2 ("b sub 2"), and h the altitude, the formula expressing this statement is

(b1 + b2)

2

6. A field in the form of a trapezoid has bases of 20 rd. and 30 rd. respectively, and an altitude of 15 rd. How many acres in the field?

7. A potato patch with two parallel sides, one 12 rd. long and the other 18 rd., and 12 rd. apart, produced 240 bu. of potatoes. Find the yield per acre.

8. Can you find or think of figures in the forms of trapezoids that must be measured by some one? If so, take the necessary measurements and measure them.

FINDING THE WIDTH OF A RECTANGLE

FINDING THE WIDTH WHEN THE LENGTH AND
AREA OF A RECTANGLE ARE KNOWN

301

Problems may arise in life in which one must find a dimension of a rectangle that, with a given dimension, will I have a given area.

1. George belongs to a corn club and is to see how much corn he can raise on just 1 acre. The field is just 20 rd. long. How wide a strip must he use?

160 ÷ 20 = 8.

Hence, a strip 8 rd. wide.

"What

By

The question is, times 20 makes 160?" dividing, we find it is 8. So he must use a strip 8 rd. wide.

2. Frank has a garden 50 ft. by 80 ft. Henry is going to lay out a garden 100 ft. long that will have the same area. How wide must he make his garden?

3. Grace has a plot in her garden 8 ft. by 20 ft. for beans. Helen is going to make her plot 30 ft. long. How wide must it be to contain the same area as Grace's garden?

4. Ralph and Charles are going to see which one can raise more shell beans from 4500 sq. ft. Ralph decided to make his bean patch 100 ft. long and Charles decided upon a patch 150 ft. long. How wide a strip must each lay out in order that each will have just 4500 sq. ft. in his plot?

5. Donald's father knows that to hold his winter's supply of coal when filled to a depth of 6 ft., he must have a bin with 96 sq. ft. of surface in the floor. He has space to make the bin but 12 ft. long. How wide will he have to make it? If he decides to make it but 10 ft. long, how wide must he make it?

6. Frank is going to fence in a garden plot that will contain just 5600 sq. ft. If he makes the garden 80 ft. long, how much fence will it take? If he makes it 140 ft. long, how much fence will it take?

7. George bought 80 yd. of netting to inclose a square garden. Ralph wants to inclose a rectangular garden of the same area that is only 45 ft. wide. How much netting will he need?

8. George wants a sail for his boat that will contain 25 sq. ft. How much canvas 3 ft. wide will he have to get?

9. Ralph built a dock for a boat landing that was 4 ft. wide and extended out 21 ft. in length. Had he made it only 3 ft. wide, the same lumber would have made a dock how long, granting that there would have been no waste in cutting?

10. A room 20 ft. wide must be how long to have the same floor space as a room 24 ft. by 30 ft.?

11. Donald's father built 200 ft. of 4-foot concrete walk about his house and grounds at the cost of 20¢ per square foot. Had he built the walk but 3 ft. wide, how much money could he have saved?

12. A window 30 in. wide and 5 ft. long will admit as much light as one 3 ft. wide of what length?

13. There should be 18 sq. ft. of floor area for each pupil in a schoolroom. A room 36 ft. long should be how wide if it has 40 pupils ?

14. A rectangle 20 in. wide must be how long to be as large as a 25-inch square?

15. How long must a rectangular field 12 rd. wide be to contain 1 acres?