Let E A C of Fig. 427 be the elevation of the pyramid, and F H K L of Fig. 428 the plan. The diagonal lines F K and L H represent the plan of the angles or hips, and G a point corresponding to the apex A of the elevation. From the apex A drop the line A B perpendicular to the base E C. Prolong E C in the direction of D, making BD equal to G F, one of the angles of the plan. Connect D and A. Then A D will be the slant hight of the article on one of the corners, and the radius of an are which will contain or circumscribe the pattern, as shown in the diagram From any center, as M, Fig. 429, with a radius equal to A D, describe an arc, as P R O S N, indefinitely and draw M P. From P, on the arc drawn, set off a chord, P R, in length equal to one of the sides of the pyramid shown in the plan. From R set off another chord, R O, in like manner, and repeat the same operation, obtaining O S and S N. Draw the lines M N, M S, M O and M R. Then M N S O R P will be the required pattern.

Fig. 437. - Elevation.

Fig. 428.-Plan.

Fig. 429. - Pattern.

The Envelope of a Square Pyramid.

PROBLEM 114. The Envelope of a Hexagonal Pyramid.

Let H G I of Fig. 430 represent the elevation of a hexagonal pyramid, of which D F C L B E of Fig. 431 is the plan. The first step is to construct a section on a line drawn from the center of the figure through one of its angles in the plan, as A B. From the center A erect A X perpendicular to A B, making it equal to the straight hight of the article, as shown in the elevation by G K. Draw the hypothenuse B X. Then X represents the apex and XB the side of a right cone, the plan of the base of which, if drawn, would circumscribe the plan of the hexagonal pyramid. From any convenient center, as X1 of Fig. 432, with X B of

Fig. 431 as radios, describe an arc indefinitely, as shown by the dotted line. Through one extremity of the arc to the center draw a line, as shown by D' X1.

Fig. 430 - Elevation.

Fig. 431.-Plan.

Fig. 432.-Pattern.

The Envelope of a Hexagonal Pyramid.

With the dividers set to a space equal to any side of the plan, as D E, commencing at D1, set off this distance on the arc six times, as shown. From the several points E1

B1 L1 in the arc thus obtained draw lines to the center, as shown by E1 X1, B1 X1, etc., which will represent the angles of the completed shape, and serve to locate the bends to be made in process of forming up. Then X1 D1 E1 B1 L1 C1 F1 D2 will be the complete pattern.