EXAMPLE. If the outside diameter of a flat ring is 9" and the inside diameter 7", what is the area of one side of the ring?

B = .7854 (81 — 49) = .7854 × 32 = 25.1328 sq. in. Ans.

Angles

Mechanics make two uses of angles: (1) to measure a circular movement, and (2) to measure a difference in direction. A circle contains 360°, and the angles at the center of the circle contain as many degrees as their corresponding arcs on the circumference.

Angle POE has as many degrees as arc PE. A right angle is measured by a quarter of the circumference of the circle, which is 90°.

The angle AOG is a right angle.

A

G

E

The angle AC, made with half the circumference of the circle, is a straight angle, and the two right angles, AOG and GOC, which it contains, are supplementary to each other. When the sum of two angles is equal to 90°, they are said to be complementary angles, and one is the complement of the other. When the sum of two angles equals 180°, they are supplementary angles, and one is said to be the supplement of the other.

The number of degrees in an angle may be measured by a protractor. The distance around a semicircular protractor is

PROTRACTOR-Semicircular, having 180°.

divided into 180 parts, each division measuring a degree. It is used by placing the center of the protractor on the vertex and the base of the protractor on the base of the angle to be

measured. Where the other side of the angle cuts the circular piece, the size of the angle may be read.

EXAMPLES

1. What is the area of a circular sheet of iron 8" in diameter?

2. What is the distance around the edge of a pulley 6" in diameter?

3. What is the area of one side of a flat iron ring 14" inside diameter and 18" outside diameter ?

4. A driving wheel of a locomotive has a wheel center of 56" in diameter; if the tires are 3" thick, what is the circumference of the wheel when finished?

5. Find the area of a section of an iron pipe which has an inside diameter of 17" and an outside diameter of 173⁄4".

6. Name the complements of angles of 30°, 45°, 65°, 70°, 85°.

7. Name the supplements of angles of 55°, 140°, 69°, 98° 44', 81° 19'.

8. What is the diameter of a wheel that is 12' 6" in circumference?

Triangles

A triangle is a plane figure bounded by three straight lines. Triangles are classified according to the relative lengths of their sides and the size of their angles.

A triangle having equal sides is called equilateral. One having two sides equal is isosceles. A triangle having no sides equal is called scalene.

If the angles of a triangle are equal, the triangle is equiangular.

If one of the angles of a triangle is a right angle, the triangle is a right triangle. In a right triangle the side opposite the right angle is called the hypotenuse and is the longest side. The other two sides of the right triangle are the legs, and are at right angles to each other.

да

EQUILATERAL ISOSCELES

SCALENE

RIGHT

KINDS OF TRIANGLES

Right Triangles

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides or legs.

If the length of the hypotenuse and one leg of a right triangle is known, the other side may be found by squaring the hypotenuse and squaring the leg, and extracting the square root of their difference.

EXAMPLE. If the hypotenuse of a right angle triangle is 30" and the base is 18", what is the altitude?

The area of a triangle may be found when the length of the three sides is given by adding the three sides together, dividing by 2, and subtracting from this sum each side separately. Multiply the four results together and find the square root of their product.

EXAMPLE. What is the area of a triangle whose sides measure 15, 16, and 17 inches, respectively?

EXAMPLE. What is the area of a triangle whose base is 17" and altitude 10"?

1. A ladder 17 ft. long standing on level ground reached to a window 12 ft. from the ground. If it is assumed that the wall is perpendicular, how far is the foot of the ladder from the base of the wall?

2. Find the area of a triangular sheet of metal having the base 81" and the height measured from the opposite angle 56". 3. Find the length of the hypotenuse of a right triangle with equal legs and having an area of 280 sq. in.

4. Find the length of a side of a right triangle with equal legs and an area of 72 sq. in.

5. Find the hypotenuse of a right triangle with a base of 8" and the altitude of 7".

6. What is the area of a triangle whose sides measure 12, 19, and 21 inches ?

7. What is the altitude of an isosceles triangle having sides 8 ft. long and a base 6 ft. long?