7. A ship sails 72 mi. on a northeast course and then 21 mi. on a southeast course. How far does this bring her from her starting point?

8. Find the diagonal of the floor of a room 24 ft. long, 18 ft. wide.

9. Find the height and area of an isosceles triangle whose base is 78 ft. and each of the other sides 89 ft.

10. Find the height and area of an equilateral triangle, each side of which measures 10 in.

11. Find how long the rafters must be for a barn 40 ft. wide, the ridgepole being 18 ft. above the level of the eaves, and the eaves projecting 2 ft. from the walls.

12. A rectangular park is 40 rd. long and 36 rd. wide. Find the length in rods of a walk between its opposite corners.

In the following find the length of the missing side:

278. A plane figure bounded by a curved line every point of which is equally distant from a point within called a center is a circle.

The curved line that bounds a circle is called the circumference.

Any straight line extending between two opposite points in the circumference and passing through the center is called a diameter.

Any straight line from the center to any point in the circumference is called a radius. A radius is one half a diameter.

In every circle the circumference very nearly equals the diameter multiplied by 3.1416. Test the accuracy of this statement by measuring the diameter and circumference of various circular objects, e.g. the bases of bottles, boxes, etc.

3.1416 may be represented by the Greek letter π (pi), the diameter by the abbreviation d, and the radius by r.

Then

π×а or πd, or 2 πr is the formula for finding any circumference.

A circle may be considered as made

up of triangles the sum of whose bases is the circumference of the circle and whose height is the radius.

Since the circumference

r

=

2 πr, the

area of a circle = 2 πr x or Tr2.

2'

The area of a circle equals the product of the circumference by half the radius, or the radius squared × 3.1416.

Written Exercise

279. What is the area of a circle

1. Whose circumference is 18.8496 ft. and radius 3 ft.?

2. Whose circumference is 23.562 ft. and diameter 7 ft.? 3. What is the length of the circumference of a 7-in. stovepipe?

4. What is the area of a circular table top that has a diameter of 2 ft. 9 in.?

5. The distance around a circular path is 157.08 ft. What is the area of the inclosed plot ?

10. Over how many square feet can a cow feed when tethered so that her head reaches 30 ft. from the stake?

11. The trunk of a tree in California has a circumference of 106 ft. What is the distance through and what is the area of a cross section ?

12. How many feet of surface has a circular mirror inside a frame that has a breadth of 3 in., and whose outside edge is the circumference of a circle whose diameter is 4 ft. 6 in. ?

13. How many square feet are there in the frame of the mirror?

14. What is the circumference of a wagon wheel that has a radius of 18 inches? How many revolutions of this wheel will cover a mile?

15. How many square feet of sheet iron are there in a piece of 6-inch stovepipe 2 ft. 6 in. long?

16. Formulate a rule for finding the area of a circle when the diameter is given.

MEASURE OF VOLUMES

280. A figure or object that has length, breadth, and thickness is called a solid.

A solid that is bounded on all sides by rectangles is called a rectangular solid.

The rectangles by which a rectangular solid is bounded are called the faces of the solid. Taken together the faces make the surface of the solid.

The lines formed by the meeting of any two faces are called the edges of the solid.

RECTANGULAR SOLID

A rectangular solid that is bounded by six equal squares is called a cube.

A cube having edges 1 inch in length is a 1-inch cube and contains 1 cubic inch. A solid having each edge 1 foot long contains a cubic

A solid having edges 1 yard long is called a cubic ———.

The standard unit of measurement for the contents of a solid is the cubic inch, from which are derived the cubic foot and the cubic yard.

The contents of a solid reckoned in units of solid measurement is called its

volume.

CUBE

The dimensions of a solid are its length, breadth, and thickness.

VOLUMES OF RECTANGULAR SOLIDS
Written Exercise

281. Illustrative Example. What is the volume of a block of concrete 4 ft. long, 3 ft. wide, and 2 ft. thick?

EXPLANATION. The area of the top face of the block is 4 x 3 x 1 sq. ft., or 12 sq. ft. The upper half of the block is 1 ft. thick. The upper half contains 12 cu. ft., i.e.4 x 3 x 1 cu. ft. = 12 cu. ft. But the whole block is 2 ft. thick. Then the whole block contains 2 x 4 × 3 cu. ft. 24 cu. ft. Ans. 24 cu. ft.

1. How many cubic feet of concrete will make a block 8 ft. long, 3 ft. wide, and 11⁄2 ft. thick ?

2. How many cubic feet of air are there in a room 35 ft. x 45 ft. x 12 ft.?

3. What is the weight of the water in a tank 26 ft. long, 16 ft. wide, and 5 ft. deep, if 1 cubic foot weighs 621 lb.?

4. What is the weight of a block of granite 8 ft. long, 13 ft. wide, and 11 ft. thick, at 165 lb. to the cubic foot?

5. What must be the length of a beam 16 in. by 22 in., to contain 115 cu. ft.?

NOTE. Divide the volume 115 by the product of the two dimensions given.

6. Formulate a rule for finding the volume of a rectangular solid when the dimensions are given.

PRISMS, CYLINDERS, PYRAMIDS, CONES, SPHERES

282. A prism is a body having two equal parallel polygons as bases and its other faces parallelograms.

TRIANGULAR
PRISM

SQUARE
PRISM

A prism is triangular, quadrangular, square, pentagonal, etc., according as its bases have three sides, four sides, five sides, etc.

The equal and parallel polygons are the bases of the prism, and the perpendicular distance between the bases is the altitude or height.

The parallelograms taken together form its convex surface.

283. A right or circular cylinder is a solid bounded by a uniformly curved surface and having for its bases circles that are parallel to each other. In this book cylinder means circular cylinder.

284. A pyramid is a solid bounded by one polygon, the base, and three or more triangles that terminate in one point at the top, called the vertex. Pyramids, like prisms, are named from their bases, as triangular, square, hexagonal, etc.

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

The slant height of a pyramid is a straight line from the vertex to the center of one side of the base.

CYLINDER

PYRAMID