The principle involved in cutting the pattern for the frustum of a cone is precisely the same as that for cutting the envelope of the cone itself. The frustum of a right cone is a shape which enters so extensively into articles of tinware that an ordinary flaring pan, an elevation and plan of which are shown in Fig. 451, has been engraved for the purpose of illustration. An inspection of the engraving will show that C D, the top of the pan, is the base of an inverted cone, its apex B being at the intersection of the lines D O and C A forming the sides of the pan; and that A D is the top of the frustum or the base of another cone, A O B. which remains after cutting the frustum from the original cone. For the pattern then proceed as follows:

Through the elevation draw a center line, K B, indefinitely. Extend one of the sides of the pan, as, for example, D O, until it meets the center line in the point B. Still greater accuracy will be insured by extending the opposite side of the pan also, as shown - the three lines meeting in the point B - which determines the apex of the cone to a certainty. Then B O and B D, respectively, are the radii of the arcs which contain the pattern. From B or any other convenient point as center, with B O as radius, strike the arc P Q indefinitely, and likewise from the same center, with B D as radius, strike the are E F indefinitely. From the center B draw a line across these arcs near one end, as P E, which will be an end of the pattern. By inspection and measurement of the plan determine in how many pieces the pan is to be constructed and divide the circumference of the pan into a corresponding number of equal parts, in this case three, as shown by K, M and L. With the dividers or spacers step off the length of one of these parts, as shown from M to L, and set off a corresponding number of spaces on the arc E F, as shown. Through the last division draw a line across the arcs toward the center B, as shown by F Q. Then P Q E E will be the pattern of one of the sections of the pan, as shown in the plan.

Fig. 451.   The Envelope of the Frustum of a Right Cone.

Fig. 451. - The Envelope of the Frustum of a Right Cone.