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3. What is a two-mile square ? What are two square miles ?
4. In what particular respect are a square yard, a square rod, and a square mile alike? In what respect are they different?
The area of a polygon is the number of square units of surface of a certain value it contains. The square inch, the square foot, the square yard, the square rod, the square chain, the acre, and the square mile are the principal English surface measure units used in computing areas.
Area of a Rectangle. The area of a rectangle is the product of its base and altitude.
1. Draw a rectangle 6 in. long and 3 in. wide. Divide the edges into parts 1 in. long, and draw lines connecting the opposite points of division, as shown in the diagram.
2. What is the shape of the parts into which you have divided the rectangle?
3. How many of these parts have you ? Count them, writing inside of each square its number from one upwards. 4. How many
with six in each row?
with three in each row ? 6. What is the area of your rectangle?
7. Can you make a rule for finding the area of a rectangle?
8. Will your rule apply to a square as well ?
1. What is the area of a field 48 rd. long and 36 rd. wide ?
2. What is the value of a square field 80 rd. long, at $371 an acre?
3. A piece of land is 70 yd. long and 60 yd. wide, and has the shape of a rectangle. Draw a plan of the field on the scale of 20 yd. to the inch, that is, let 1 inch represent 20 yd. on your plan. What is the area of your plan? What is the area of the field ?
4. Given the area and one side of a rectangle, how do you find the other side ?
5. What is the length of a field whose width is 60 rd. and whose area is 5400 sq. rd ?
Area of a Rhomboid. The area of a rhomboid is equal to the product of its base and altitude.
As the area of a rectangle is equal to the product of its base and altitude, this is also the area of the rhomboid.
LESSON 111 1. What is the area of a rhomboid whose length is 72 rd. and width 48 rd.?
2. A piece of land is 800 rd. long and 60 rd. wide, and has the shape of a rhomboid. What is its value at $50 an acre ?
3. A room 15 ft. long requires 20 sq yd. of carpet to cover the floor. How wide is the room?
4. What is the area of a rhomboid whose length is 9 chains and width 5 chains? Express the area : 1st, in square chains; 2d, in square feet; 3d, in square rods; 4th, in acres.
5. The area of a room 36 ft. long is 120 sq. yd. What is the width of the room ?
Area of a Rhombus. As a rhombus is a rhomboid whose four sides are of equal Rhombus length, the area of a rhombus is equal to the product of the base and altitude.
6. How many square rods are there in a field in the form of a rhombus, each side measuring 64 rods, and the perpendicular between opposite sides 50 rods? How many acres ?
7. The base of a rhombus is 30 yd. and altitude 70 ft. Find its area.
8. The base of a rhomboid is 30 ch. and the altitude 25 rods. What is its area?
9. What is the area of the walls of a room 30 ft. long, 242 ft. wide, and 14 ft. high ? What is the area of the ceiling?
10. Find the cost of plastering the walls and ceiling of the room described in Ex. 9, at 20% a square yard.
Area of a Triangle. The area of a triangle is equal to one half of the product of its base and altitude.
The Base is any one of its sides.
B В ular distance from the base to
D the vertex of the opposite angle. Thus, in the triangle ABC, the base is AB, and the altitude CD.
1. Draw on paper a parallelogram ABCD of any convenient size as shown in the diagram. Taking AB for the base, draw the
с altitude DE, and the diagonal DB. Cut out the parallelogram from the paper, and cut it into two parts along the diagonal DB. Now turn one part A
E around and place it directly on top of the other, and you will see that the two triangles are equal. There are several kinds of triangles; but all can be formed by cutting quadrilaterals into two parts from corner to corner.
As the area of a parallelogram is equal to the product of its base and altitude, so the area of a triangle, which is one half the parallelogram, is one half the product of the base and altitude of the parallelogram, that is, one half the product of its own base and altitude.
2. What is the area of a triangle whose base is 24 ft. and altitude 14 ft.?
3. The base of a triangle is 20 rd. and the altitude is 12 rd. What is the area of the triangle?
4. Find the area of a triangle whose base is 24 yd. and altitude 2 rd.
5. How many acres are there in a triangular piece of land having a base of 80 rd. and an altitude of 56 rd.?
Draw the following triangles on a scale of } in. to the foot, and calculate their areas. Write the area and the value of the given parts in and about the diagrams:
Altitude 3 ft.
1. Base 6 ft. and altitude 3 ft. 2. Base 4 ft. and altitude 2 ft. 3. Base 8 ft. and altitude 4 ft. 4. Base 5 ft. and altitude 3 ft.
5. A right triangle, the sides of the right angle being 5 ft. and 4 ft.
6. A right isosceles triangle whose equal sides are 5 feet.
Area of a Trapezoid. The area of a trapezoid is equal to its altitude multiplied by one half of the sum of the parallel sides.