PROPOSITION XV. THEOREM 784. The circumferences of two circles are to each other as their radii, and the circles are to each other as the squares of their radii. Let A and B be two circles and R and r be their radii. To Prove Proof. Inscribe similar regular polygons in the two circles. Let P and p denote the perimeters of these polygons. As the number of sides is indefinitely increased, P and p approach circumference A and circumference B respectively as their limits. (?) The members of equation (1) may therefore be regarded as two variables that are always equal, and since each is approaching a limit, their limits are equal. (?) Circumf. B r 785. COROLLARY I. The circumferences of two circles are to each other as their diameters, and the circles are to each other as the squares of their diameters. 786. COROLLARY II. The ratio of the circumference of a circle to its diameter is constant; that is, it is the same for all circles. The value of this constant is denoted by the Greek letter π. i.e. The circumference of a circle is π times its diameter. If, in the formula for the area of a circle, area= circumf. × R, the value of the circumference just derived is substituted, we obtain area= πR2. i.e. The area of a circle is π times the square of its radius. 787. DEFINITION. angles at the center. Similar arcs are arcs that subtend equal Since the intercepted arcs are the measures of the angles at the center, similar arcs contain the same number of degrees of arc, and are consequently like parts of their circumferences. Similar sectors are sectors the radii of which include equal angles, or intercept similar arcs. Similar segments are segments whose arcs are similar. 788. COROLLARY III. Similar arcs are to each other as their radii. [See definition.] 789. COROLLARY IV. Similar sectors are to each other as the squares of their radii. [§§ 776 and 788.] 790. COROLLARY V. the squares of their radii. Similar segments are to each other as 791. EXERCISE. The circumferences of two circles are 942.48 ft. and 157.08 ft. respectively. The diameter of the first is 300 ft. Find the diameter of the second. 792. EXERCISE. What is the ratio of the areas of the two circles of the preceding exercise ? 793. EXERCISE. How many units in the radius of a circle, the area and circumference of which can be expressed by the same number? PROPOSITION XVI. PROBLEM 794. To find an approximate value of T. The perimeter of a circumscribed square (see § 729) is 4 D (D = diameter). P' The perimeter of an inscribed square is 2√2 D=2.8284271 D. Substituting 4 D for P and 2.8284271 D for p in the formulas 2 px P (1) and p' = √p × P' (2), we get P' or the perime P+P ter of the circumscribed octagon = 3.3137085 D, and p' or the perimeter of the inscribed octagon 3.0614675 D. = Substituting 3.3137085 D for P and 3.0614675 D for p in formulas (1) and (2), we obtain values for the perimeters of the circumscribed and the inscribed polygons of sixteen sides. Substituting these values, the perimeters of polygons of thirty-two sides are obtained. SANDERS' GEOM.-16 Continuing in this way, the following table is formed: The circumference of the circle therefore lies between 3.1415926 D and 3.1415928 D. For ordinary accuracy the value of π is taken as 3.1416. NOTE. -The value of T has been carried out over seven hundred decimal places. [See article on " 'Squaring the Circle" in the Encyclopædia Britannica.] The value of to thirty-five decimal places is 3.14159265358979323846264338327950288. By higher mathematics, the diameter and circumference of the circle have been shown to be incommensurable, so no exact expression for their ratio can be obtained. 795. EXERCISE. The radius of a circle is 10 in. Find its circumference and its area. 796. EXERCISE. The area of a circle is 7854 sq. ft.. Find its circumference. 797. EXERCISE. The circumference of a circle is 50 in. What is its area? 798. EXERCISE. The radius of a circle is 50 ft. a sector whose arc contains 40° ? What is the area of 799. EXERCISE. The of that circle is 120 sq. ft. radius of a circle is 10 ft. The area of a sector EXERCISES 1. In a regular polygon of n sides, diagonals are drawn from one vertex. What angles do they make with each other? 2. Show that the altitude of an inscribed equilateral triangle is of the diameter, and that the altitude of a circumscribed equilateral triangle is 3 times the radius. 3. The radii of two circles are 4 in. and 6 in. respectively. How do their areas compare? 4. Find the area of the ring between the circumferences of two concentric circles the radii of which are a and b respectively. 5. The area of a regular inscribed hexagon is a mean proportional between the areas of the inscribed and the circumscribed equilateral triangles. [See Ex. to Prop. 6.] 6. The diagonals joining the alternate vertices of a regular hexagon form by their intersection a regular hexagon having an area one third of that of the original hexagon. 7. Find the area of the six-pointed star in the figure of Exercise 6 in terms of the radius of the circle. E 8. From any point within a regular polygon of n sides, perpendiculars are drawn to the sides. Prove that the sum of these perpendiculars is equal to n times the apothem of the polygon. [Join the point with the vertices and obtain an expression for the area of the polygon. Compare this with the expression for the area obtained from § 771.] 9. Construct a circle that shall be double a given circle (§ 784). 10. Construct a circle that shall be one half a given circle. 11. Construct a circle equivalent to the sum of two given circles; also one equivalent to their difference. [§ 646.] |