107. Measurement of a Rectangle. If this figure represents a rectangle 4 in. long by 2 in. wide, there are evidently 4 sq. in. in the lower row of squares, and since there are 2 rows, there are 2 × 4 sq. in., or 8 sq. in., in the rectangle. If we know the area and one dimension, we may find the other dimension; for, since 8 sq. in.

= 2x4 sq. in., then 8 sq. in. ÷ 4 sq. in. = number of units in the other dimension.

2, the

Express the length and breadth of a rectangle in the same linear unit; the product of these two numbers will express the area in square units of the same name as the linear unit.

The number of square units in a rectangle divided by the number of linear units in one dimension gives the number of linear units in the other dimension.

This means that the numbers are to be considered as abstract. The complete solutions with concrete numbers are given above.

The expression 4" x 7" means and is read 4 inches by 7 inches.

EXERCISE 55

Find the areas of rectangles with these bases and altitudes:

1. 127 ft., 46 ft.

2. 152 in., 35 in.

3. 225 yd., 82 yd.

4. 240 in., 11 ft.

5. 6 ft. 4 in., 3 ft.

6. 7 ft. 2 in., 5 in.

7. 3 ft. 6 in., 1 ft. 4 in.
8. 9 yd. 2 ft., 3 ft. 6 in.

9. How many square rods in a rectangular field 32 rd. long and 16 rd. wide?

10. How many acres in a rectangular field 126 rd. long and 38 rd. wide?

11. A rectangular city lot is 1291 ft. long and 25 ft. 9 in. wide. Find the number of square feet in the lot.

12. A rectangular field containing 4.05 A. is 36 rd. long. How wide is it?

108. Triangle. A plane figure bounded by three straight lines is called a triangle.

The perpendicular distance from the vertex to the line of the base is called the altitude or height of the triangle.

109. Parallelogram. A plane figure bounded by two pairs of parallel lines is called a parallelogram.

The base of a parallelogram is the side on which it rests. The altitude is the perpendicular distance from the opposite side to the base. The opposite parallel sides of a parallelogram are equal.

the par

If we cut off the triangle T and place it at X, allelogram will be changed to a rectangle of the same base and the same altitude.

Therefore, the area of a parallel

ogram is equal to the product of the

base and altitude.

X

For example, if the base of a parallelogram is 3 ft. 8 in. and the altitude is 3 ft., what is the area?

3 x 3 sq. ft. 11 sq. ft.

If the area of a parallelogram is 15 sq. ft. and the base is 5 ft., required the altitude.

110. Measurement of a Triangle. We see that this triangle may be cut into two parts, each of which is half of some rectangle, and that therefore the whole tri

angle is half of a rectangle of the same base

and altitude.

Whatever the shape of the triangle, it may, in a similar way, be shown to be half of a parallelogram of the same base and altitude, and therefore (§ 109) equal to half of a rectangle of the same base and altitude.

Therefore, the area of a triangle equals half the product of the numbers representing the base and altitude.

EXERCISE 57

Find the areas of triangles with the following bases and altitudes

15. At $5.50 a square foot, what is the value of a triangular city lot with base 82 ft. and altitude 45 ft. 6 in.?

16. What is the area of a triangle with sides all equal, whose perimeter is 18 in. and altitude 5.19 in. ?

17. A triangular gable to a house has a base 30 ft. and an altitude 10 ft. 6 in. Find the area of the gable.

18. What is the altitude of a triangle with area 37 sq. in. and base in. ?

19. What is the altitude of a triangle with area 7 sq. in. and base 31 in.?

20. What is the base of a triangle with area 47 sq. ft. and altitude 23 ft.?

21. The gable of a house is a triangle with base 32 ft. and altitude 9 ft. What is the area of the gable?

22. In laying out a triangular flower bed the sides are all made equal. The perimeter is found to be 33 ft. and the altitude 9.5 ft. What is the area?

23. The four faces of a church spire are triangles, each with base 24 ft. and altitude 53 ft. How many square feet in each triangle? in the whole surface of the spire?

24. A man buys a 54-acre lot that is a parallelogram in shape. The frontage along the road is 144 rd. What is the distance to the rear line?

25. For the panels of a staircase 20 marble slabs are needed. They are parallelograms, each with base 1 ft. 6 in. and altitude 4 ft. 3 in. What is the total area of all the slabs?

26. A marble floor is made of 975 equilateral triangles, each with base 10 in. and altitude 8.6 in. What is the area of the floor space in square inches, making no allowance for

cement?

27. A corner lot, where two streets meet at oblique angles, is a parallelogram in shape. Its length along one street is 63 ft., and the shortest distance from that front to the side parallel to it is 98 ft. What is the area?

28. A triangle, a square, a rectangle, and a parallelogram all have the same base and the same altitude. The side of the square is 44 in. What is the area of the square? of the triangle? of the rectangle? of the parallelogram?