PROBLEM VI. 347. To find the numerical value of, approximately. Let C be the circumference, and R the radius of a O. established in (346), we make the following computations: In a regular inscribed polygon of 24. BC V2—√ 4—(.51763809)2= .26105238 6.26525722. 48. BC V2-1 4—(.26105238)2= .13080626 6.27870041. 96. BC=√2—√ 4—(.13080626)2= .06543817 6.28206396. 192. BC V2-14—(.06543817)2= .03272346 6.28290510. 384. BC=√2—√/4—(.03272346)2= .01636228 6.28311544. 768. BC=√2-14-(.01636228)2= .00818121 6.28316941. It will be seen that the first four decimal places remain the same, to whatever extent we increase the number of sides. Hence we can consider 6.28317 as the approximate value of the circumference of a circle whose radius is 1. EXERCISES IN INVENTION. THEOREMS. 1. The side of an inscribed equilateral triangle equals half the side of the circumscribed equilateral triangle. 2. The diameter of a circle is a mean proportional between the sides of the equilateral triangle and the regular hexagon circumscribed about the circle. 3. The square inscribed in a circle equals half the square on the diameter. 4. The area of a regular inscribed hexagon equals threefourths the area of a regular circumscribed hexagon. 5. The area of a regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles. 6. If the vertices of a square are taken as centres and half the diagonal as a radius and circles be described, the points of intersection of the circumferences and the sides of the square are the vertices of a regular octagon. 7. The area of a regular inscribed octagon equals the area of a rectangle whose adjacent sides equal the sides of the circumscribed and inscribed squares. 8. The area of a regular inscribed dodecagon equals three times the square on the radius. PROBLEMS. 1. Inscribe in a given circle a regular polygon similar to a given regular polygon. 2. Circumscribe a polygon similar to a given inscribed polygon. 3. In a given circle, inscribe three equal circles, touching each other and the given circle. 4. In a given circle, inscribe four equal circles in mutual contact with each other and the given circle. 5. In a given equilateral triangle, inscribe three equal circles, touching each other, and each touching two sides of the triangle. 6. About a given circle, describe six circles, each equal to the given one and in mutual contact with each other and the given circle. |