5. Over how many square feet of surface can a horse graze if it is fastened to a post with a rope that allows it to graze 100 feet from the post?

R 8 2

6. A circular garden having a radius of 8' is divided into quadrants by cross walks 2' wide. It is also bounded by a circular walk 2' wide. How many bricks are required to lay the walks when the bricks, 21" by 4" by 81", are laid flat? Regard the cross walks as rectangles 16' long and 2' wide.

(Allow 39 bricks to 1 sq. yd.)

Indicating Radius by R, Diameter by D, Circumference by C, and Area by A, the two rules given on p. 421 may be expressed by formulas 1, 2, and 3 below.

Since the square of any number is the square of twice the number,

the square of the radius is the square of the diameter, or R2

D2
4

=

D2

4

Substituting for R2 in formula 3, and canceling the factor 4, you get,

4. D2x.7854=A.

From the foregoing formulas, other formulas may be derived by applying the principle: If the product of two factors is divided by either factor, the quotient is the other factor.

Formulas 1 and 2 and 3 should be thoroughly memorized; the remaining formulas may be derived from these when needed.

Find the required element in the following exercises and express the formula employed for finding it:

What is

19. The area of a circular garden is 2578.23

20. The circumference of a circle is 15.708 ft.

its area? How many circles, each 15 in. in diameter, will equal the area of the above circle?

21. A hot air flue is 24 in. by 36 in. What is the diameter of a circular flue having the same area?

22. What is the difference in area of two flues, one 16 in. square, and the other a circular flue 16 in. in diameter ?

23. Draw a figure to represent an 8-in. square and circumscribe a circle about it. What is the difference in the

areas of the two figures?

24. The diameter of a water pipe is 3 in. What is the area of the cross section of the pipe?

25. A circular silo has an inside diameter of 20 ft. How many square feet are there in the base of the silo?

SOLIDS

A solid or body has length, breadth, and thickness. A prism is a solid whose sides are parallelograms, and whose two ends are equal polygons parallel to each other. Prisms, according to the form of their bases, are triangular, quadrangular, pentagonal, etc.

A triangular prism is one that has triangles for its bases. A quadrangular prism is one that has quadrilaterals for its bases. Quadrangular prisms are also called parallelopipeds.

A cube is a solid with six equal square faces.

A cylinder is a solid with a curved surface and two equal parallel circular bases.

The convex surface of a cylinder is the area of its curved surface.

The axis of a cylinder is the line joining the center of the two bases.

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

CYLINDER

A pyramid is a solid whose base is a polygon, and whose sides are triangles meeting at the vertex.

The altitude of a pyramid is the distance from the vertex to the center of the base; as AB in the figure below.

The slant height of a pyramid is the distance from the vertex to the middle of the base of any of its triangular sides; as AC in the figure.

A cone is a solid whose base is a circle and whose surface tapers uniformly to a point.

A sphere is a solid with a curved surface, every point of which is equally distant from a point within called the center.

The diameter of a sphere is a straight line passing through the center of the sphere and terminating in its surface. One half the diameter is the radius of the sphere. The greatest distance around the sphere is its circumference.

All the surface of a solid except its base or bases is called the lateral surface. The entire surface is the sum of the lateral surface and of the bases.

STUDY RECITATION

NOTE. Derive the rules for finding the area of the surface of different solids in manner similar to the one that follows:

C

B

Cover exactly the lateral surface of any cylinder with a sheet of paper of the same height. Unroll the paper. Find its area. What is the circumference of the cylinder? Multiply the circumference of the cylinder by its height. How does the product compare with

the area of the paper? State the rule for finding the lateral surface of a cylinder.

The cylinder in the illustration is represented as lying on a plane touching it along the line AB. Conceive it rolled to the right. When A reaches the plane at A', B is exactly at the point B', and the cylinder has made one rotation. The path de

A

B

Α'

B'

scribed by it is a rectangle equal to the entire lateral surface of the cylinder.

The sides of any prism are rectangles. If these rectangles are placed side by side, they form one rectangle whose length is the perimeter of the base and whose width is the altitude of the prism.

The surface of a cylinder may be regarded as made up of an infinite number of rectangles, the sum of whose bases is the circumference of the cylinder. The length of each rectangle is the altitude of the cylinder.

The

The lateral surface of a pyramid is made up of triangles. The slant height of the pyramid is the altitude of each of the triangles. perimeter of the base is the sum of the bases of the triangles.

A cone may be regarded as a pyramid with an infinite number of bases, therefore its lateral surface is made up of an infinite number of triangles whose altitude is the slant height of the cone and the sum of whose bases is the circumference of the base of the cone.

In finding the lateral surface of a prism, a cylinder, a pyramid or a cone, you simply find the area of three or more rectangles, or three or more triangles.