traced to the base of the cone, then to the miter line. With a radius equal to the distance from this point to the apex, and with the apex as a center, a curved extension line intersecting lines 1 and 1 of the stretchout is drawn. From point 2 of the profile, the extension line is traced to the base of the cone, then to the miter line, and thence horizontally to the slant height line. With a radius equal to the distance from this point to the apex, and with the apex as a center, a curved extension line intersecting lines 2 and 2 of the stretchout is drawn. In like manner, the remaining intersections of the stretchout may be traced. A curved line passing through these points will give the miter cut of the roof flange at the roof line. The upper miter line, being parallel to the base, is developed like an ordinary right cone. With a radius equal to the slant height, a curved extension line passing through the stretchout is first drawn. The necessary allowances for locks parallel to lines 1 of the stretchout are added. One-half inch double edges to the upper and lower miter cuts of the pattern are also added to allow for joining to the pipe and apron.
43. Related Mathematics on Conical Roof Flange.—Area of Frustum.—If a right cone is cut by a plane parallel to that of the base, the top section will still be a right cone although of small dimensions, and the lower part will be a frustum of a cone. The profiles of both bases, or ends, will be circles. The smaller circle is generally called the upper base, and the larger circle the lower base of the frustum. The lateral area of a frustum of a cone is found by adding together the circumferences of the upper and lower bases, dividing the sum by 2, and then multiplying by the slant height. This is often expressed as a formula for area of a frustum:
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A frustum whose lower base has a circumference of 40" and
