To describe an equilateral triangle.

From the points A, B, as centres, and with A B as radius, describe arcs intersecting each other in C. Draw C A, C B, and the figure A B C will be the triangle required. Fig. 7.

To describe a square.

On

From the point B, draw B C perpendicular, and equal to A B. A, and C, with the radius A B, describe arcs cutting each other in D. Draw the lines DA, DC, and the figure ABCD will be the square required. Fig. 8.

To inscribe a square in a circle.

Draw the diameters A B, C D perpendicular to each other. Then draw the lines A D, A C, B D, BC; and ABCD will be the square required. Fig. 9.

To inscribe an octagon in a circle.

Bisect any two arcs AC, BC of the square A B C D in G, and E. Through the points G, and E, and the centre o draw lines, which produce to F, and H. Join AF, FD, DH, &c., and they will form the octagon required. Fig. 9.

On a line to describe all the several polygons, from the hexagon to the dodecagon.

Bisect A B by the perpendicular CD. From A as a centre, and with A B as a radius, describe the arc B E, which divide into six equal parts; and from E as a centre describe the arcs 5 F, 4G, 3 H, &c. Then from the intersection E as a centre, and with E A as a radius, describe the circle AID B, which will contain A B six times. From F in like manner as a centre, and with F A as radius, describe the circle A KL B, which will contain A B seven times; and so on for the other polygons. Fig. 10.

To inscribe in a circle an equilateral triangle.

From any point D in the circumference as a centre, and with the radius DO of the given circle, describe an arc A O B cutting the circumference in A, and B. Through D, and o draw D C. Then, join A B, AC, BC; and the figure ABC will be the triangle required.

Fig. 11.

To inscribe a hexagon in a circle.

Bisect the arcs A C, B C in E, and F, and join A D, D B, B F, &c., which will form the hexagon. Or carry the radius six times round the circumference, and the hexagon will be obtained. Fig. 11.

To inscribe a dodecagon in a circle.

Bisect the arc A D of the hexagon in G, and AG being carried twelve times round the circumference, will form the dodecagon. Fig. 11.

To inscribe a pentagon, hexagon, or decagon, in a circle. Draw the diameter A B, and make the radius D C perpendicular to A B. Bisect DB in E. From E as a centre, and with E C as radius, describe an arc cutting A D in F. Join CF, which will be the side of the pentagon, CD that of the hexagon, and D F that of the decagon. Fig. 12.

To find the angles at the centre, and circumference of a regular

polygon.

Divide 360 by the number of the sides of the given polygon, and the quotient will be the angle at the centre; and this angle being subtracted from 180°, the difference will be the angle, at the circumference, required.

Table, showing the angles at the centre, and circumference.

To inscribe any regular polygon in a circle.

From the centre C draw the radii C A, C B, making an angle equal to that at the centre of the proposed polygon, as contained in the preceding table. Then the distance A B will be one side of the polygon, which, being carried round the circumference the proper number of times, will complete the polygon required. Fig. 13.

To circumscribe a circle about a triangle.

Bisect any two of the given sides, A B, B C by the perpendiculars E F, D f. From the intersection F as a centre, and with the distance of any of the angles, as a radius, describe the circle required. Fig. 14.

To circumscribe a circle about a square.

Draw the two diagonals A C, B D intersecting each other in o. From O as a centre, and with O A, or O B, as a radius, describe the required circle. Fig. 15.

To circumscribe a square about a circle.

Draw the two diameters A B, C D perpendicular to each other, through the points A, C, B, D, draw the tangents E F, E G, G H, F H, and EG HF will be the square required. Fig. 16.