Parallelograms are quadrilaterals with opposite sides equal and parallel to each other, therefore the opposite angles are equal, the diagonals bisect the figures into similar triangles and also bisect each other. They are simple to construct as shown, where 341:4 is a square, 341:5 a rectangle, 341:6 a rhombus and 341:7 a rhomboid.



342 Polygons


Regular polygons—a pentagon has five sides (342:2A), a hexagon six sides (IB), a heptagon seven sides (1C), an octagon eight sides (1D), etc.—have more than four sides, the sides are equal and the angles also equal. All the corners must, therefore, lie on the circumference of the


circumscribing circle, and given the circle it is only necessary to divide the circumference into equal parts representing the number of sides required. Alternatively, any regular polygon can be drawn on a given base-line AB (342:2) by erecting a perpendicular BC equal in length to AB and joining C to A. AB is then bisected at Y
and a perpendicular XY of indefinite length erected which bisects the diagonal AC to yield the centre of a square raised on AB. If a quadrant (quarter-circle) is now drawn with centre B and radius AB, then where it cuts the perpendicular XY will lie point 6 which is the centre of a circle circumscribing a hexagon, while midway between points 4 and 6 the point 5 will give the centre of a pentagon, and similarly placed points 7, 8, 9 and 10, etc. the centres of other polygons. Other methods of drawing regular polygons are also shown. In 342:3 a hexagon on the given side AB is simply drawn with a 60° set square forming six equilateral triangles, while 342:4 illustrates the construction of an octagon within a given square ABCD. The diagonals of the square are first drawn, then with radius equal to half a diagonal arcs are drawn from centres A, B, C, D giving points of intersection with when joined become the sides of the octagon. Figure 342:5 shows the construction of a regular polygon within a given circle. The circle is first drawn with diameter AB, and a semicircle of any convenient radius with centre A. If this semicircle is divided into as many parts as the required polygon is to have sides—in this case a heptagon—then radials extended from A through these points will cut the circumference and give points of connections for the base-lines of the required figure.

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