Page:Sheet Metal Drafting.djvu/46

18. Related Mathematics on Chimney Tube.—Circumference of a Circle.—A circle is a portion of a flat surface bounded by a curved line, every point in which is the same distance from a point within, called the center. The circumference is simply the curved line that is drawn with the compass. The diameter of a circle is any straight line that passes through the center of the circle and has its ends in the circumference. It is possible to draw any number of diameters in the same circle, but they will all have the same length.

Value of $\scriptstyle {\pi }$.—The girth or distance around a cylinder can be found by wrapping a narrow strip of newspaper around it. The point where the strip overlaps the end should be marked before the paper is removed from the cylinder. The distance from the end of the paper to the mark will give the distance around the cylinder, or the girth.

If the diameter of the cylinder be accurately measured and the circumference or girth divided by the diameter, the answer will be about $\scriptscriptstyle {3{\frac {1}{7}}}$. Regardless of the size of the cylinder this experiment will always produce the same result. Mathematicians have proved that the exact relation of circumference to diameter cannot be found. The value 3.1416 is near enough for most purposes. Some sheet metal workers use $\scriptscriptstyle {3{\frac {1}{7}}}$ or $\scriptscriptstyle {\frac {22}{7}}$ in their computations. This relation between circumference and diameter is indicated by the Greek letter $\scriptstyle {\pi }$ (pronounced pi).

Suppose it is desired to find the circumference of a 7" circle: $\scriptstyle {7''\times {\frac {22}{7}}={\frac {154}{7}}=22''}$, or 7″ × 3.1416 = 21.9912″. If the circumference is given and the diameter is wanted, the process is reversed; i.e., with a circumference of 26″, $\scriptstyle {26''\div {\frac {22}{7}}=26''\times {\frac {7}{22}}=8{\frac {3}{11}}''}$; or $\scriptstyle {26''\div 3.1416=8.2442''}$.

Lateral Area of a Cylinder.—Lateral means pertaining to the side. Lateral area is the area of the side wall of a cylinder. The pattern of the side wall of a cylinder is a rectangle whose length is equal to the circumference of the profile, and whose height is the height of the cylinder. The area of the pattern is equal to the length times the height; therefore, the lateral area of any cylinder must be equal to the circumference of the base times the height.

Problem 5A.—Compute the circumference of the chimney tube, Fig. 42, and compare the answer to the length of the line of stretchout between points 1 and 1. These should agree or a mistake has been made.