WRITTEN EXERCISES.

1. If a ball 4 in. in diameter weighs 10 lb., what does a ball 12 in. in diameter weigh?

Since the cubes of the like dimensions of similar solids are to each other as their volumes, then 43, the diameter of the first cubed, is to 123, the diameter of the second cubed, as 10 lb., the weight of the first, is to the From which we find the weight of the larger

2. If a ball 5 in. in diameter weighs 24 lb., what is the diameter of a similar ball which weighs 81 lb.?

3. If a stack of hay 8 ft. in diameter weighs 15 tons, what is the diameter of a similar stack which weighs 120 tons?

4. If a globe of gold 2 inches in diameter is worth $500, what is the value of a globe 6 inches in diameter?

5. The diameters of two similar cisterns are to each other as 1 to 8; what is the relation of their contents?

6. If a bin 10 ft. long holds 50 bushels, how long is a similar bin which holds 400 bushels?

7. If a bin 15 ft. long and 10 ft. wide contains 800 cubic feet, what are the dimensions of a similar bin which contains 2700 cubic feet?

8. If a bar of metal 3 ft. long, 2 ft. wide, and 1 ft. thick weighs 2700 lb., what are the dimensions of a similar bar which weighs 6400 lb.?

9. If the surfaces of two spheres are to each other as 4 to 9, what is the relation of their contents?

10. If the volumes of two spheres are to each other as 250 to 128, what is the relation of their surfaces?

11. If a log 3 ft. in diameter contains 70 cubic feet, what is the diameter of a log of the same length that contains 210 cubic feet?

MENSURATION.

235. Mensuration is the process of computing the lengths of lines, the areas of surfaces, and the volumes of solids.

SECTION XI.

LINES.

236. A Line is that which has length only.

Straight Line.

Curved Lines.

A Straight Line is a line that does not change its direction. A Curved Line is a line that changes its direction at every point.

Parallel Lines are lines which have the same direction.

D

ANGLES.

237. An Angle is the figure formed by two straight lines which meet at a point; as, CDA, CDB, EFG, HKL.

E

Right Angles.

B F

Acute Angle.

Parallel Lines.

G Κ

Obtuse Angle.

When one straight line meets another, making the adjacent angles equal, the lines are perpendicular to each other, and the angles are Right Angles.

An Acute Angle is an angle less than a right angle. An Obtuse Angle is an angle greater than a right angle.

SURFACES.

238. A Surface is that which has length and breadth only. A Plane Surface, or a Plane, is a surface in which, if any two points are taken, the straight line which joins these points will lie wholly in the surface.

239. A Plane Figure is a plane surface bounded by lines either straight or curved.

240. A Polygon is a plane figure bounded by straight lines.

Triangle.

Quadrilateral.

241. A Polygon of three sides is called a Triangle; one of four sides, a Quadrilateral; one of five sides, a Pentagon; one of six sides, a Hexagon, etc.

Pentagon.

The lines A B, BC, CA; DE, EF, FG, GD; and HI, IJ, JK, KL, LH, of the above polygons are called the sides, and in any polygon the sum of the sides is called the perimeter.

The diagonal of a polygon is a line joining any two vertices not consecutive.

The area of a plane figure is the number of square units it contains.

242. A Triangle is a polygon of three sides; as, A BC.

The base of a triangle is the side upon which it seems to stand; as, AB. The point opposite the base is called the vertex; as, C.

The altitude of a triangle is the perpendicular drawn from the vertex to the base; as, CD.

A4

Triangle.

E

243. A Parallelogram is a quadrilateral whose opposite sides are parallel; as, ABCD.

C

D

Parallelogram.

E

A Rectangle is a parallelogram whose angles are all right angles; as, A B C D.

B

G

Trapezoid.

Rectangle.

Square.

A Square is a rectangle whose sides are all equal; as, EFGH.

A Trapezoid is a quadrilateral

two of whose sides are parallel;
as, M N OP. The two parallel
sides are called the bases of the
trapezoid; as, MN and PO. The M
altitude is the perpendicular dis-
tance between the bases; as, PQ.

N

AREA OF POLYGONS.

244. The Rectangle.

Let the figure ABCD represent a rectangle 8 inches long What is its area?

and 4 inches wide.

If we divide the base and altitude into inches, and draw lines through these points of division parallel to the sides, the rectangle will be divided into There will be 8 equal squares. square inches in one row; and since there are 4 such rows, there will be 4 times 8 square inches, or 32 square A

inches.

From this we derive the

D

B

RULE.

To find the area of a rectangle, multiply the base by the altitude.

WRITTEN EXERCISES.

1. What is the area of a rectangle 20 ft. long and 16 ft. wide?

2. What is the area of a rectangle 12 ch. long and 5 ch. wide?

3. How many square yards in a rectangle 40 ft. long and 18 ft. wide?

4. How many acres in a rectangle 40 rd. long and 12 rd. wide?

5. How many acres in a rectangle 36 chains long and 10 chains wide?

6. Find the area of a square whose sides are each 8 ft. 6 in.

7. Find the length in rods of a rectangular field whose area is 16 acres and width 40 rd.