283. Similar Surfaces.

1. Similar surfaces are such as have the same shape, such as circles, squares, right-angled triangles, etc.

2. Two similar surfaces are to each other as the squares of like dimensions.

EXERCISES.

284. 1. A triangular lot whose base is 8 rds. has an area of 15 sq. rds. What is the area of a similar lot whose base is 12 rds.?

STATEMENT:

rds.

rds. sq.rds. sq.rds. 82 122 :: 15 : ?

2. The diameters of two circles are to each other as 2 to 3. If the smaller one contains 20 sq. in., how many square inches in the larger one?

3. A circle whose diameter is 3 ft. is

what part of a circle whose diameter is 4 ft.? 6 ft.? 7 ft.? 9 ft.?

4. How many circles 10 in. in diameter contain the same area as a circle 20 in. in diameter?

5. A circle whose diameter is 3 ft. is how many times a circle whose diameter is 3 in.? 6 in. ? 9 in.?

6. The areas of two similar triangles are respectively 32 in. and 50 in. The base of one is 8 in. What is the base of the other?

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If the area of an equilateral triangle is 18 sq. ft., what is the area of a similar triangle, each of whose sides is twice as long? 3 times as long? 7 times as long?

CYLINDER.

PYRAMID.

CONE.

measure. Volume.

2. A prism is a solid, two of whose faces, called ends or bases, are equal, parallel, and similar polygons, and the rest of whose faces are rectangles.

3. A cylinder is a solid whose bases are equal and parallel circles, and whose sides are perpendicular to the bases.

4. The altitude of a prism or a cylinder is the perpendicular distance between its bases.

5. A pyramid is a solid whose base is a polygon and whose other faces are triangles meeting in a point called the vertex.

6. A cone is a pyramid whose base is a circle.

SPHERE.

7. The altitude of a pyramid or of a cone is the perpendicular distance from its vertex to its base.

8. A sphere is a solid, all points of whose surface are equally distant from a point within called the center.

9. The volume of any prism or cylinder equals the product of its base and its altitude. (See Section 149.)

10. The volume of a pyramid or a cone is one third of the volume of a prism or a cylinder having the same base and altitude.

11. The surface of a sphere is equal to 3.1416 times the square of its diameter.

12. The volume of a sphere is equal to 3.1416 times one sixth of the cube of the diameter:

EXERCISES.

286. 1. How many cubic feet in a prism whose base is a rectangle 5 ft. long and 4 ft. wide, and whose altitude is 13 ft.? How many in a pyramid of the same dimensions?

2. What is the volume of a triangular prism whose altitude is 15 ft., and each side of whose base is 4 ft.? (See Problem 11, Section 280.) Of a pyramid of the same dimensions?

3. What is the volume of a cylinder the diameter of whose base is 12 in. and whose altitude is 27 in.? Of a cone having the same base and altitude?

4. How many cubic feet in a circular granite column whose circumference is 3 ft. and whose altitude is 16 ft.? In a cone having the same base and altitude?

5. Find the surface and volume of a sphere whose diameter is 18 in.

6. Find the surface and volume of a sphere whose circumference is 18 in.

287. Similar Solids.

1. Similar solids are such as have the same shape.

2. The volumes of similar solids are to each other as the cubes of their like dimensions.

EXERCISES.

288. 1. The sides of two cubes are to each other as 3 to 5. If the smaller one contains 27 cu. in., what is the volume of the larger?

2. A marble whose diameter is 1 in. is what part of a marble whose diameter is 2 in.? 3 in.? 4 in.?

3. A sphere whose radius is 2 in. is what part of a sphere whose radius is 4 in.? 6 in.? 9 in.?

4. How many baseballs 3 in. in diameter equal in volume a football 9 in. in diameter?

5. Which would you rather have, 4 oranges 3 in. in diameter, or 2 oranges 4 in. in diameter? Why?

PROGRESSIONS.

289. Write the number 3. Now write a number two larger than 3, then a number two larger than 5, and so on, till you have written six numbers. Write 50, then 47, then 44, and so on till you have written six numbers.

Write the number 2. Now write a number three times as great, then a number three times as great as the second, and so on till you have written six numbers. Now write 729; then write a number one third as great, and so on till you have written six numbers.

You have now written four series of numbers, for when several numbers increase or diminish by a constant difference, or by a constant ratio, they constitute a series; they are said to be in Progression. You will notice that there are two kinds of progression; the first is called arithmetical progression, and the second is called geometrical progression. The common difference may be a whole number or a fraction; and the same is true of the common ratio.

If the numbers making the series are constantly growing larger, the series is an Ascending series; if they are constantly growing smaller, they constitute a Descending series. Of course, by reversing the order of the terms, any ascending series will become a descending series, and vice versa.

The first term of a series and the last are the extremes; all the other terms are means.