The length of an arc cannot be developed accurately by geometrical means, but for all practical purposes the two following methods will be found adequate. In Fig. 1, let A a 1? be the arc whose length is required. Draw the chord B A and produce it to C, making A C half the length of B A. From C, with the radius C B, draw part of a circle, and from A draw the tangent A D, cutting this circle in the point P. Then the line A D will be approximately equal in length to A B, being a trifle short of the real length. If the arc subtends an angle of 60°, the error will be about one-thousandth part of the length. The second method is more accurate, giving results a trifle full. Let A B in Fig. 2 be the arc whose length is required, and C the centre of the circle of which it forms a part. Bisect the arc in D, and bisect D A in E. Draw C E and produce it. From A draw the tangent A F, cutting C E produced in the point P. Draw the straight line B F. Then a straight line of the length A F + F B will be approximately equal in length to the arc AB. Apart from geometrical construction, the length of the arc may be measured by stepping a pair of dividers round the arc, counting the number of steps taken, and then setting out the same number of steps along a straight line. This will always give a result short of the actual length, but the smaller the opening of the dividers the more accurate will be the result. A more accurate way is to use a wheelinefna, or a special instrument called an opisometer. The length of the arc may be calculated as follows. Set out the arc either full size or to as large a scale as possible, as in Fig. 3. Measure the chord AB, bisect it, and set up a perpendicular cutting the arc in C. Measure A C, which is the chord of half the arc. The length of the arc is found by multiplying the length of AC, the chord of half the arc, by 8, from this product subtracting the length of the chord A B, and dividing the remainder by 3. If the radius of the curve is known, and also the number of degrees contained in the angle (V),the length of the arc may be calculated in another way, as follows. The circumference of the whole circle is found by multiplying twice the radius by 3-1416. Then, as the circumference contains 360°, the length of the arc will be proportionate to the number of degrees it contains, and can be arrived at by a simple rule of three sum, thus, 360J: degrees in the arc:: circumference: length of arc.

Developing Length of Arc.