To Lay Out Regular Polygons With A Steel Square. (A.) Any regular polygon, or any polygon of equal sides and angles, may be inscribed within a circle. Each side of such a polygon forms the base of an imaginary triangle, the apex of which is the center, or axis of the polygon, and the sides of which form the miter of the polygon. See the construction of a hexagon, Fig. 50.

Thus each of the five triangles, which constitute a fivesided polygon or pentagon, will have 360°/ 5 = 72° in its vertex angle.

The angles formed by the base of each triangle, or side of the pentagon with the sides of its triangle, may be found by dividing the vertex angle by two, and subtracting the quotient from 90°; in the case of a pentagon, this angle would be, 90° - 72° / 2 = 54°.

In using a steel square, to lay out a right angle triangle, the student should remember that the sum of the two angles with the hypotenuse is always 90°, or the angle of a perfect square, because this is the mathematical basis of nearly all of the work which can be done with the steel square. Thus, if one angle of a right-angled triangle is 40° with its hypotenuse, the other angle will be 50°, and if the blade of the square is at an angle of 30° with the diameter of the semicircle in Fig. 48, the tongue will be at an angle of 60°. If pins were driven at a and b, and the square rotated against them, the angle c would describe the arc of a circle.

Fig. 47. - Dividing a Board into Equal Spaces.

Note. - Hereafter "the tongue" or "on the tongue" will be designated by To., and " the blade " or "on the blade," by Bl.

In laying out polygons, the student should construct the geometric figure, extending the base line as in Fig. 50, in order to find the exact figures on To. and on Bl., which will construct the angle. Numbers commonly used will be remembered easily.

Hold the square as described in Topic 31, B; keep 12 Bl. upon the base line, and swing the square around until To. exactly coincides with the side of the polygon. (B.) An equilateral triangle (Fig. 49), constructed in this way, will give 12 Bl., 6 11/12 To.; To. = outside angle of the sides as at a. The miter of the triangle will be found upon the blade at b. Throughout work of this kind, the number 12 will be almost invariably the blade number, hence the tongue number will be the only one to be kept in mind.

Fig. 48. - Construction of a Circle with a Steel Square.

Fig. 49. - Construction of an Equilateral Triangle.

Note. - The figures given may not be found absolutely correct if calculated mathematically, but they are sufficiently accurate for all practical purposes, and adapted to the 1" scale upon the square.

(C.) The hexagon may be constructed by joining six equilateral triangles within a circle, the radius of which equals one side of a triangle. The vertex angles of a hexagon are 360°/ 6 = 60°; thus the same figures upon the square are used as for the equilateral triangle, the tongue giving the angle, as at a, Fig. 50. The miter, being the side of an equilateral triangle, is found by using the same figures, the tongue giving the miter cut, as at b.

Fig. 50. - Construction of a Hexagon.

Fig. 51. - Construction of a Rectangle.

(D.) The sides of a rectangle are square with the base line, or at an angle of 90°, the miter of which, 90°/2 = 45°, may be obtained by using the same figures on both To. and Bl., say 12, either side giving the cut. (Fig. 51.)

(E.) The octagon, or eight-sided polygon, treated in the same way, gives 12 To., 12 Bl., for the outside angle, either side giving the cut. For the miter, 12 Bl.,4 11/12 To.; To. = cut. (See Fig. 52.) The above-mentioned polygons are the ones commonly used by carpenters, though any regular figure may. be constructed by the same methods.

(F.) A bevel set to the figures upon the square which have been found by the above methods will be found more convenient than the square itself in marking cuts and angles, if a number are to be made alike.