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99. As an exercise we will calculate the length of the side of an equilateral triangle inscribed in a circle, and also the length of the side of a regular polygon of twelve sides.

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Draw the straight lines FB, BD, DF; an equilateral triangle.

B

K

thus we form

Suppose the radius of the circle 1 inch required FB. This is an example of Art. 95. Let O be the centre of the circle; draw OA cutting BF at K.

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square root of 3: proceeding to seven places of decimals we obtain BF=1.7320508.

Thus the side of the equilateral triangle inscribed in the circle is 1.7320508 inches.

Again, let OL be perpendicular to AF, and produce it to meet the circumference again at M. Join AM. Then AM is one of the sides of a regular polygon of twelve sides inscribed in the circle. We can calculate AM by Art. 93.

1

AL L=1, OA=1; thus OL= of the square root of 3

=8660254: therefore LM=1339746. Then AM is the square root of 2679492, which we shall find to be 51764 very nearly.

Thus the side of a regular polygon of twelve sides inscribed in the circle is 51764 inches very nearly.

EXAMPLES. VII.

1. The height of an arc is 15 inches, and the chord of half the arc is 4 feet 6 inches: find the diameter of the circle.

2. The height of an arc is 2:28 feet, and the chord of half the arc is 7-15 feet: find the diameter of the circle.

3. The chord of half an arc is 3 feet 4 inches, and the diameter of the circle is 25 feet: find the height of the arc.

4. The chord of half an arc is 6:43 feet, and the diameter of the circle is 23.65 feet: find the height of the arc.

5. The height of an arc is 1 foot 3 inches, and the diameter of the circle is 11 feet 3 inches: find the chord of half the arc.

6. The height of an arc is 3.24 feet, and the diameter of the circle is 28.76 feet: find the chord of half the arc.

7. The chord of an arc is 20 feet, and the height of the arc is 4 feet: find the diameter of the circle.

8. The chord of an arc is 15'78 feet, and the height of the arc is 2.8 feet: find the diameter of the circle.

9. The chord of an arc is 15 inches, and the diameter of the circle is 20 inches: find the chord of half the arc.

10. The chord of an arc is 80 inches, and the diameter of the circle is 100 inches: find the chord of half the arc.

11. The chord of half an arc is 2 feet 6 inches, and the diameter of the circle is 4 feet 2 inches: find the chord of the arc.

12. The chord of half an arc is 24 feet, and the diameter of the circle is 16 feet: find the chord of the arc.

13. The chord of an arc is 12 yards, and the chord of half the arc is 19 feet 6 inches: find the diameter of the circle.

14. The chord of an arc is 49 feet, and the chord of half the arc is 25 feet: find the diameter of the circle.

VIII. CIRCUMFERENCE OF A CIRCLE.

100. We often require to know the proportion which the length of the circumference of a circle bears to the length of the diameter: the proportion cannot indeed be stated exactly, but it can be stated with sufficient accuracy for any practical purpose.

101. The diameter of a circle being given, to find the circumference.

RULE. Multiply the diameter by 34, that is by

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other words, multiply the diameter by 22, and divide the product by 7.

102. Examples.

(1) The diameter of a circle is 4 feet 8 inches.

4 feet 8 inches=56 inches,

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Thus the circumference is about 176 inches, that is,

about 14 feet 8 inches.

(2) The diameter of a circle is 4.256 feet.

4'256

22

8512

8512

7 93.632

13.376

Thus the circumference is about 13.376 feet.

103. The Rule of Art. 101 makes the circumference a little greater than it ought to be. The circumference of a circle is in fact less than 30 times the diameter, but greater than 3 times. The rule of multiplying the diameter by 34 is generally found sufficiently accurate in practice.

104. We may if we please put the Rule of Art. 101 in the form of a proportion, and say, as 7 is to 22 so is the diameter of any circle to the circumference.

105. The following proportion is still more accurate: as 113 is to 355 so is the diameter of any circle to the circumference. This Rule also makes the circumference a little greater than it ought to be; but the error is excessively small, being at the rate of rather less than a foot in nineteen hundred miles.

106. We may also put the proportion in the following form: the diameter of any circle is to the circumference as 1 is to 3.141592653589793...; the calculation of this proportion has been carried to more than 600 places of decimals. We may use as many as we please of the figures which have been obtained: it is very common to take 3*1416 as a sufficient approximation.

107. Accordingly the Rule for finding the circumference of a circle when the diameter is given, may be stated thus: multiply the diameter by 37; or, if greater accuracy is required, multiply the diameter by 3·1416.

The latter form of the Rule also makes the circumference a little greater than it ought to be; but the error will not be so much as part of the circumference: so that the error will be at the rate of less than a foot in seventy-five miles.

1 400000

108. When we are told to multiply the diameter by 31416, we may, if we please, multiply 31416 by the diameter. A similar remark applies to all rules relating to the multiplication of numbers.

109. Examples.

(1) The diameter of a circle is 42.7 inches.

3.1 416

4 2.7

219912

62832

1256 64

13414632

Thus the circumference is nearly 134.14632 inches.

(2) The diameter of a circle is 8000 miles.

3.1416

8000

25132 8000

Thus the circumference is nearly 25132.8 miles.

110. The beginner should exercise himself in actually measuring the diameter and circumference of some circle, as for example, a wheel. Although he may not be able to obtain very accurate results, yet he may convince himself that the circumference is about 34 times the diameter.

111. The circumference of a circle being given, to find the diameter.

22

RULE. Divide the circumference by 37, that is by ; 7 in other words, multiply the circumference by 7, and divide the product by 22. Or, if greater accuracy is required, divide the circumference by 3.1416.

112. Examples.

(1) The circumference of a circle is 50 feet.

50

7

2350

11175

1 5.9

Thus the diameter is about 15'9 feet.

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