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angle; nevertheless, custom has sanctioned a different unit. The unit of angle generally adopted is an angle equal toth part of a right angle, called a degree, and denoted by the symbol. The corresponding unit of arc is th part of a quadrant, and is also called a degree. A right angle and a quadrant are both expressed by 90°. Two right angles and a semi-circumference are both expressed by 180°. Four right angles and a whole circumference are both expressed by 360°.

The degree either of angle or arc is subdivided into minutes and seconds, denoted by the symbols' and "; a minute being the th part of a degree, and a second theth part of a minute. Fractional parts of a degree less than a second are expressed by decimal parts of a second.

An angle, or an arc of any magnitude, is then numerically expressed by the unit degree and its subdivisions. Thus, for example, an angle equal to 4th of a right angle, as well as its intercepted arc, will be expressed by 12° 51′ 25′′.714....

Definition 1.-Two angles are complemental when their sum is 90°.

Def. 2.-Two angles are supplemental when their sum is 180°.

PROP. XIV. To find the number of degrees, minutes, &c. in an angle at the centre of a circle subtended by an arc equal to the radius.

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This is the magnitude of the angle in any circle which is subtended by an arc equal to the radius, and as an arc equal to the radius is taken for the unit arc, so this angle is taken for the unit angle, and serves for measuring the magnitude of other angles.

Numerical Examples.

1. What is the magnitude of that angle whose circular measure is? Or, in other words, what is the magnitude of that angle which is subtended by an arc equal to of the radius?

It has been shewn above that an angle subtended by an arc equal to the radius is

57° 17′ 45′′;

therefore that subtended by an arc equal to the radius will be

28° 38′ 52′′.

2. What is the circular measure of an angle of 72°? The circular measure of 180° is ;

.. 180° 72° : : : circular measure of 72°;

.. circular measure of 72°

2

72

T= - T.

180

Remark.-It has been shewn that

T= = 3.14159;

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and it should be borne in mind that is always equal to this number; but sometimes the young student may find taken to represent 180°, and in that case he may be reminded that, still retaining its original signification, is understood to mean an angle which is 3.14159 times the unit angle (57° 17′ 45′′).

Exercises (3).

1. Find the circumference of a circle whose radius is

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3. Find the circumference of a 4-inch drain pipe.

4. The difference between the circumference and diameter of a circle is

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Find the radius of the circle in both cases.

5. The sum of the circumference and diameter of a circle is

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Find the radius of the circle in both cases.

6. A circular grass-plot measures 50 feet in diameter; find its circumference.

7. Find the circumference of a circle whose radius is 12 feet 9 inches.

8. How often will a carriage wheel which is 3 feet 6 inches in diameter turn round in going a distance of I mile?

9. A wheel which is a foot and a half in diameter revolves 1200 times; through what distance has it passed?

10. What must be the radius of a wheel which revolves 220 times in passing over one half-mile?

II. Find the circumference described by the point of the minute-hand of a clock 33 inches in length.

12. Compare the rates at which the points of the hour and minute hands of a watch move, their lengths being respectively and of an inch.

13. Compare the perimeter of a square, and that of the inscribed and circumscribed circles.

14. Compare the circumference of a circle with the perimeters of the inscribed and circumscribed hexagons.

15. The earth moving round the sun at the rate of 1135 miles per minute, takes 365 days to complete her revolution; shew how to find the distance of the earth from the sun, the orbit being supposed circular.

16. Find the least of the two angles subtended by the hands of a watch at

(1.) 25 minutes past 12 o'clock.

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17. When between 3 and 4 o'clock will the hands subtend an angle of 45°?

18. In a circle whose radius is 10 inches, find

(1.) The angle subtended at the centre by an arc of 15 inches.

(2.) The angle subtended at the centre by an arc of 20 inches.

(3.) The angle subtended at the centre by an arc of I foot.

19. Find also the corresponding angles subtended at the circumference.

20. In a circle whose diameter is 40 feet, find the arc of 25° 30' at the centre.

21. What must be the radius of a circle, so that an angle of 75° at the centre is subtended by an arc of 36 inches?

22. In a circle whose radius is 1 foot, find the angle subtended by an arc of 1 inch.

23. Find the angle at the circumference which is subtended by an arc of 10 inches in a circle whose radius is 10 inches.

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24. Find the number of degrees in an angle formed by two chords intersecting:

(1.) Within the circle,

(2.) Without the circle;

and intercepting arcs equal to m and n, the radius of the circle being r.

25. Find the radius when an angle of 25° at the circumference has an arc of 100 inches.

26. Find the angle whose circular measure is

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27. Find the circular measure of the following angles : 20°, 45°, 108°, 235°, 112° 30′, 150°, 300°.

28. Compare the radii of two circles, in which an angle of 40° at the centre of the one and an angle of 60° at the centre of the other have equal arcs.

29. Find the complements of the following angles : 25°, 37° 15′,

42° 25′ 15′′.

75°, 81° 10′, 7° 15'.

30. Find the supplements of the following angles : 110° 15', 71°, 108° 25'.

25° 10', 60°, 135°

31. The circular measure of an angle is ; find

(1.) Circular measure of the complement.

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32. What is the magnitude of that angle, the circular measure of whose supplement is three times the circular measure of its complement?

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