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MEASURE OF ANGLES. 1. Angular Unit.—2. Ratio of Circumference to Diameter.
3. Division of the Circle.
1. Angular unit.-An angle is defined by geometers to be the inclination of one right line to another.
In order to express this kind of magnitude by numbers, it is necessary to select an angular unit, to which all other angles may be referred. As this selection is altogether arbitrary, different units may be proposed: in this Chapter it is not necessary to consider more than those two, which are usually employed.
The angular unit which is commonly used in mathematical treatises may be thus defined:
DEFINITION: The angular unit* is that angle at the centre of a circle which is subtended by an arc equal in length to the radius.
Thus if the arc AU (fig. 1) be taken equal in length to the radius CA, the angle ACU will be equal to the angular unit. Any other angle ACB may be expressed numerically, with reference to this unit, as follows:
Let N be the number which expresses its value; let a be the length of the arc AB, which subtends it; and r the radius of the circle. The following proportion is evident:
L ACB: LACU:: arc AB : arc AU.
* For an account of the other angular unit, vid. p. 5.
But as ACU is the angular unit, and AU equal to the radius, it follows that
N:1::a:1, and therefore N =
(1) That is, The numerical value of an angle, is the number which expresses the ratio of the arc which subtends it, to the radius of the circle.
In equation (1) three quantities, viz. N, a, r, are connected together, from which it follows, that if any two be given, the third may be calculated.
1. If the radius of a circle be 35 feet, calculate the angle at the centre, which is subtended by an arc of 6 feet. Having reduced the fraction to the decimal form, we find
Ans. 0,17142. 2. If the radius be 12 feet 7 inches, and if the arc be 5 inches, find the angle at the centre.
Ans. 0.03311. 3. If the radius be 97 feet, and the angle at the centre = 0.734, calculate the length of the subtending arc.
By equation (i), it is evident that 97 must be multiplied by 0734, in order to obtain the result.
Ans. 71.198 feet. 4. If the radius be 10 feet 9 inches, and the angle at the centre 0.01347, calculate the arc.
Ans. 1.73763 inches. 5. An angle at the centre 1.25 is subtended by an arc whose length is 16 feet, calculate the radius.
In this case it is evident from equation (1) that 16 must be divided by 1.25 to obtain the result.
Ans. 12.8 feet. 6. If the angle at the centre be 0.00157, and the subtending arc 6 inches, calculate the radius. Ans. 318 feet, 5.65 inches.
2. Ratio of circumference to diameter.-The ratio of the circumference of a circle to its diameter has been determined by geometers to be as follows: