This section is from the book "The Building Trades Pocketbook", by International Correspondence Schools. Also available from Amazon: Building Trades Pocketbook: a Handy Manual of reference on Building Construction.
In making hoppers or boxes with inclined sides, it is necessary to obtain the face and edge bevels; when the sides are mitered, the edge bevel is called the miter bevel, and When they are butted, it is termed the butt bevel. In Fig.25 the diagrams are assumed to be laid out on a board, as if in bench practice.
In (a), let c b be the width of a side of the hopper, and c e the splay; let c be on the edge k I. Strike the arc bj, and draw cj square to k I; then cj is the width of the stuff. Lay out the thickness b a; draw a d parallel to c b; also, c d parallel to b a; and cbad will represent a section of the board. Project b to b', and draw c b'; then the angle g b'c will be the face bevel. Make h h' equal to b a, and draw g' h'; project a to a'; and draw b'a', then gb'a' will be the miter bevel. If the edges are cut off along the line fa, the miter bevel will be 45°.
If the hopper is to have butt joints, the face bevel is found as above shown To obtain the butt bevel, proceed as follows: Draw af through a, parallel to g h; project f to f" on g' h', and draw b'f; then the angle g b'f ' will be the butt bevel.
In (b) is shown a perspective view of the board kg hl, with the bevels in position, gb'c and gb'a' being those required for a miter-jointed hopper, and g b'c and g b'f ' those for a butt-jointed one. At c"f ' b' c is shown an outline of the butt joint, and at c c"'a' b' that of the miter joint.