It is important for the student of mechanical drawing to become thoroughly familiar with many fundamental geometric figures, because they play a vital part in all branches of the subject. Most nuts, for example, are either hexagonal or octagonal, and a knowledge of regular polygons is necessary to facilitate the drawing of these machine parts. The same is true for cylinders, cones, pyramids, triangles and quadrilaterals. Undoubtedly you have already become acquainted with these geometric figures in your elementary high school geometry, but a review here will help to refresh your memory and show you their application to mechanical drawing.

ANGLES

On the opposite page you will see a number of geometric figures. They are all numbered for the purpose of reference. Numbers 1, 2, and 3 show three kinds of angles. The acute angle shown in 1 is any angle that is less than 90 degrees; the right angle in 2 is, of course, always 90 degrees and can never be anything else; the obtuse angle in 3 is always greater than 90 degrees. The method of drawing these angles has already been described under the heading "Protractor," so that it is unnecessary to dwell on their construction.

TRIANGLES

In Figures 4, 5, 6 and 7 we see the different kinds of triangles. The scalene triangle in 4 is the general term for all triangles, while the right triangle (5), the equilateral triangle (6), and the isosceles triangle (7) are special forms of the scalene. Any triangle, one of whose angles is 90 degrees, is called a right triangle; any triangle, all of whose sides are equal, is called an equilateral triangle; any triangle, two of whose sides are equal, is called an isosceles triangle. Since the sum of all the angles in any triangle is always equal to 180 degrees, it is obvious that no triangle can contain more than one right angle. The three angles of an equilateral triangle are all equal, so that each angle must equal 60 degrees. In an isosceles triangle the two base angles are always equal to each other. The method of constructing triangles depends, of course, upon the kind of triangle and the parts that are given. If you are given three sides of a triangle, the simplest method of constructing it is to lay off one of the sides on a horizontal line and, with the extremities of this line as centers and the lengths of the other two lines as radii, draw two arcs. (See the top of page 36.) The point where the two arcs meet will be the apex of the required triangle. The right triangle may be constructed with either of your two triangles, the horizontal and vertical sides being measured and the hypotenuse drawn to complete the triangle. The equilateral triangle may be drawn with your 60-degree triangle.

QUADRILATERALS

Any four-sided figure is a quadrilateral. Just as a scalene triangle usually refers to any triangle whose sides and angles are not necessarily equal, so a quadrilateral refers to any four-sided figure whose sides are not necessarily equal. Number 16 shows a quadrilateral in general, and numbers 8 to 12 inclusive show special kinds of quadrilaterals. The square (8) and the rectangle (9) are similar figures. They are defined as any quadrilateral, all of whose four angles are right angles. The square is an equilateral rectangle— a rectangle all of whose sides are equal. The rectangle is not a square, since two of its sides are unequal in length to the other two sides. The rhomboid (10) is any quadrilateral whose opposite sides are equal and parallel but none of whose angles is 90 degrees; while the rhombus is an equilateral rhomboid—all its sides are equal but none of its angles is a right angle. The ace of diamonds is an example of a red rhombus.

Now all of the quadrilaterals, from 8 to 11 inclusive, have something in common: their opposite sides are all equal and parallel. For this reason they are classed as parallelograms. A parallelogram is any quadrilateral whose opposite sides are equal and parallel. The trapezoid (12) is not a parallelogram because only two of its sides are parallel and no one side is necessarily equal to any other side. Any quadrilateral, two of whose sides are parallel, is a trapezoid. One fundamental and important rule of all parallelograms is that the diagonals (the lines drawn from the opposite corners), meet in the center. To get the center of any of the figures from 8 to 11 inclusive, simply draw the two diagonals; and where they cross each other will be the center of the figure. Most of the drawing that you do will be done on rectangular sheets of paper. To determine the center of the paper it is entirely unnecessary to do any measuring. Merely draw the two diagonals and mark a light cross where the diagonals meet.

Another interesting and very important application of geometry to drawing is shown by the fact that the diagonal of any rectangle, if produced, will give other rectangles which are proportional to the original one. This is demonstrated in Figure 17 at the bottom of page 36. The value of this application becomes apparent when you want to enlarge or reduce a drawing or a picture. If, for example, you have a drawing that is 2 5/16 inches long and 1 inch high, and you want to make it 3 15/16 inches long, you can easily determine its height without doing a lot of unnecessary arithmetic. Just lay off the original rectangle with the 2 5/16-inch side horizontal, then draw the diagonal and extend it. Now measure 3 15/16 inches along the horizontal and, with the triangle, draw a vertical line up ta the diagonal. The length of this line will be the length of the side of the new rectangle. This is shown in Figure 17.

REGULAR POLYGONS

The prefix poly means many; the suffix gon means side. A polygon is therefore a figure of many sides. A regular polygon is a polygon all of whose sides are equal in length. Only regular polygons will be considered here.

The name polygon usually applies to figures of five sides or more. A regular polygon of five sides is a regular PENTAGON (13); one with six sides is a regular HEXAGON (14); one with seven sides is a regular HEPTAGON; one with eight sides is a regular OCTAGON (15), and so on.

If you recall your elementary geometry you will remember that the number of straight angles (180-degree angles) in any polygon is always two less than the number of sides in that polygon. A regular pentagon has five sides so that it must have three straight angles, or 540 degrees. Each angle is therefore one-fifth of 540 degrees, or 108 degrees. A regular hexagon has 4 straight angles, or 720 degrees, and each angle is therefore 120 degrees. In a similar way you may determine the number of degrees in any regular polygon.

Regular polygons may be thought of as making equal triangles. Five equal triangles make a regular pentagon, six make a hexagon, seven a heptagon and so on. Since there are 360 degrees in a circle, the angle at the center must always be 360 degrees divided by the number of triangles. The angles at the center of the triangles in a regular pentagon are therefore 72 degrees or one-fifth of 360 degrees. In a regular hexagon they are 60 degrees, or one-sixth of 360, in an octagon, 45 degrees or one-eighth of 360, and so on.

You can readily see that as the number of sides of a regular polygon increases, the perimeter of the polygon approaches the circumference of the circle in which it is inscribed. It is therefore best, in drawing any regular polygon, to draw the circle first and place the polygon in it.

Suppose it is required to construct a regular pentagon. The first thing to do is to draw a circle, and through the center draw a vertical and horizontal line dividing the circle into four quadrants (Figure 16, page 36). Now bisect the right half of the horizontal diameter in O and, with O as a center and a radius equal to O A, draw an arc cutting the horizontal diameter in B. The length AB is equal to one side of the regular pentagon; this length should then be laid off on the circumference with the dividers. The angle at the center must be 72 degrees, or 360 divided by 5.

The construction of the hexagon and octagon are particularly easy. In the hexagon, one side equals the radius, so that, after you have drawn the circle, mark off the radius on the circumference as shown in Figure 14, page 36. Do this six times and you will have a regular hexagon. The regular hexagon may also be drawn with the 60-degree triangle, since it contains six equilateral triangles all of whose angles are 60 degrees each. In the same way, an octagon may be drawn with the 45-degree triangle (Figure 15). A simple method of dividing a circle into any number of equal parts or, in other words, of drawing a regular polygon of any number of sides, is given in Figure 22 on page 40. The description of this method, as well as the description of other constructions, is given on that page.

SOLIDS

The solids, 17 to 27 inclusive, are divided into parallelopipeds (cubes, solid rectangles and solid parallelograms), cylinders, cones, prisms and pryamids. Since they are all three dimensional, they can be drawn only in perspective and cannot be constructed in the same manner that plane figures can. The solids all speak for themselves and there is no need to dwell on them at length here. The cylinder and right cone are the two most important, the cylinder because it is so common in tools and machinery, and the cone because of its sections.

CONIC SECTIONS

In Figure 18 we see four right cones, each cut in a different way. In a, the cone is cut parallel to the base and the resulting section is a circle. In b it is cut slantingly and the section is an ellipse. In c it is cut parallel to the side of the cone and the section is a parabola; in d it is cut at right angles to the base and the section is a hyperbola. All these four sections are known as CONIC SECTIONS of which the circle and the ellipse are the most important in mechanical drawing.

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