MEASUREMENT-PYRAMIDS

Instruments: Rule, compasses and triangle

1. Construct an equilateral triangle of sides 3 in.

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2. Bisect the sides of the triangle and connect these points by lines, as shown in Fig

ure 1.

3. Fold the triangle along these lines and bring the corners together.

A

Fig. 1.

Fig. 2.

4. Construct a square of side 2 in. On each side of the square construct an isosceles triangle of side 3 in. (See Fig. 2.)

5. Fold so as to bring the vertices of the triangles together.

Pyramids. The forms made in Exercises 3 and 5 are called pyramids.

The base, or bottom, of a pyramid may have any number of sides. Pyramids are called triangular, square, and the like, according to the shape of the base. What kind of figures are the other faces of a pyramid?

Fig. 3.

John wrapped a piece of paper around a square pyramid and trimmed it off even with the base. He also wrapped a piece of paper around a Fig. 4. square prism with base and altitude equal to those of the pyramid, leaving one end open and trimmed off even with the solid. After pasting the margins so as to hold the forms firmly in shape, he filled the paper pyramid with sand and then poured the sand into the paper prism. By doing this three times, he filled the prism.

6. If convenient, perform an experiment similar to this.

7. The volume of the pyramid was how many times as great as that of the prism? How did their bases and altitudes compare? How is the volume of a square prism found?

8. According to the result of John's experiment, how may the volume of a square pyramid be found?

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MEASUREMENT-PYRAMIDS

Chester drew a square 1 in. on a side and a triangle of equal area, as shown in the figure. He modeled from clay two pyramids of equal altitudes, one having the

square for the base and the other the triangle. The weights of the two pyramids were the same. Chester concluded that the volume of a triangular pyramid may be calculated in the same way as that of a square pyramid.

1. If Chester's conclusion was correct, how may the volume of a triangular pyramid be calculated when the altitude and the area of the base are known?

It can be proved that: The volume of any pyramid, no matter what its base, is of the product of the area of the base and the length of the altitude.

2. Find the volume of a triangular pyramid, the area of whose base is 36 sq. in. and whose altitude is 8 in.

3. The pyramid of Cheops, shown in the picture, stands on a square base 746 ft. on a side; what is the area of

puted? Find the lateral area (surface, excluding the base) of the pyramid of Cheops (altitude of side 608 ft.).

5. Find the numbers to fill the blanks:

Oral.

REVIEW

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1. What is an equilateral triangle? An isosceles triangle?

2. Explain how to construct with the aid of compasses an equilateral triangle of side 4 in.; also explain how to construct an isosceles triangle of base 3 in. and sides 4 in.

3. If a circle is drawn through the vertices of a regular hexagon, how does the length of each side of the hexagon compare with the length of the radius of the circle?

4. Name some object in the form of a pyramid. What shape may the base of a pyramid have? The other sides?

5. How is the volume of a pyramid computed? What is the volume of a pyramid of base 16 sq. in. and altitude 51⁄2 in. ? Written.

6. Draw a square of side 2 in. Draw its diagonals. With the corners of the square as centers describe arcs as shown in the figure. Complete the figure.

7. A pyramid is 50 ft. square and 75 ft. high. Find its volume.

8. The Washington monument is crowned with a square pyramid as shown in the picture. The vertex of the pyramid is 25 ft. above its base. Find its volume.

9. Find the altitude of a pyramid of which the volume is 84 cu. ft. and the base is a square 6 ft. on a side.

Find the missing numbers in this table concerning pyramids :

Oral.

EQUATIONS

LITERAL NOTATION

1. If t represents the age of a man now, what represents his age 10 years from now? y years from now? 12 years ago? n years ago?

2. If A earns b dollars in a week and spends c dollars in the same time, what represents the amount that he saves?

3. A piece of merchandise is bought for d dollars and sold at $10 profit; what stands for the selling price?

4. A watch was sold for $25 and there was $d profit; what stands for the cost of the watch?

5. A box and contents weigh 140 lb. The box weighs p pounds. What stands for the weight of the contents?

6. If one book weighs p pounds and another q pounds, what is their combined weight?

7. At x cents a peck, what represents the cost of 1 bushel?

8. At b cents an ounce, what represents the cost of 3 oz.? Of 1 lb.? Of 6 lb.? Of p lb.?

4a

a

a

a

9. If 5 represents the side of a b square, what represents its perimeter? 10. How many dimes are there in 5 dollars? In d dollars? In 3x dollars?

b

11. Find the perimeter of the figure at the left.

12. How many ounces are there in 3 pounds? In x pounds? In 5a pounds?

Oral.

THE BALANCE

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1. How heavy must the weight, w, be in order that the two pans may balance each

other?

The condition of balanc

ing may be represented by

w+2=7.

State the condition of balancing for each of the following:

2.

3.

4.

5. The picture shows a balance made by suspending a square piece of board from each end of a yardstick.

A

2 lb

quart measure filled with wheat is placed on one board and enough weights to balance it on the other.

If the cup when empty weighs 4 oz., what does the wheat in the cup weigh? If the wheat is poured loosely on one side of the balance, how many ounces are needed on the other side to balance it?

The solution may be outlined thus:

1. Let w be the number of ounces in the weight of the wheat. 2. Then, w+4= the number of ounces in the weight of the wheat and the cup.

3. Then, w + 4 = 34. Why? 4. Hence, w=30. Why?

Written.

6. A half-peck measure weighs lb. When filled with wheat, it weighs 81 lb. Find the weight of pk. of wheat. Express the solution in the form given in Exercise 5.

7. Find the value of w in Exercises 1, 2, 3, and 4.