10. A circle is the limit, a. Of a regular inscribed polygon when the number of its sides is indefinitely increased. § 392 b. Of a regular circumscribed polygon when the number of its sides is indefinitely increased. 11. Figures are in proportion, § 392 a. If they are regular polygons of the same number of sides, to the squares upon their radii. $ 390 b. If they are regular polygons of the same number of sides, to the squares upon their apothems. c. If they are circles, to the squares of their radii. § 390 d. If they are similar sectors, to the squares of their radii. § 399 e. If they are similar segments, to the squares of their radii. § 401 $ 402 12. The area of a figure is equal, b. If it is a circle, to one half the product of its circumference by its radius. a. If it is a regular polygon, to one half the product of its perimeter by its apothem. § 389 c. If it is a circle, to times the square of its radius. § 397 § 398 d. If it is a sector, to one half the product of its arc by its radius. § 400 SUPPLEMENTARY EXERCISES Ex. 688. If the perimeter of each of the figures, equilateral triangle, square, and circle is 396 ft., what is the area of each figure ? Ex. 689. If the inscribed and circumscribed circles of a triangle are concentric, the triangle is equilateral. Ex. 690. If an equilateral triangle is inscribed in a circle, any side will cut off one fourth of the diameter from the opposite vertex. Ex. 691. The square inscribed in a circle is equivalent to one half the square circumscribed about that circle. Ex. 692. A circle is inscribed in a square whose side is 4 in. How much of the area of the square is without the circle ? Ex. 693. What is the width of the ring between the circumferences of two concentric circles whose circumferences are 48 ft. and 36 ft. respectively? Ex. 694. Of all squares that can be inscribed in a given square, the minimum has its vertices at the middle points of the sides. Ex. 695. Every equiangular polygon circumscribed about a circle is regular. Ex. 696. In any regular polygon of an even number of sides, the lines joining opposite vertices are diameters of the circumscribing circle. Ex. 697. Given the side of a regular inscribed polygon and the side of a similar circumscribed polygon, to compute the perimeters of the regular inscribed and circumscribed polygons of double the number of sides. Data: AB, the side of a regular inscribed polygon, and CD, the side of a similar circum- C, scribed polygon, tangent to the arc AB at its middle point E. Denote the perimeters of these polygons by P and respectively, and the number of sides in each by n; denote the perimeters of the inscribed and circumscribed polygons which have 2 n sides by S and T respectively. Required to compute the value of S and of T. F E G D Solution. Through A and B draw tangents to meet CD in F and G respectively; also draw AE and BE. Then, § 376, AE and FG are sides of the polygons whose perimeters are S and T respectively. and substituting for AE, AB, and EF their values, T ; 4 n Ex. 698. To compute the approximate ratio of a circumference to its diameter. Solution. If the diameter of a circle is 1, the side of a circumscribed square is 1, and its perimeter is 4; the side of an inscribed square is√2, and its perimeter is 2 √2, or 2.82843. Thus, Q4, and P=2 √2 for the octagon. T-2Q × P S=√PXT Q+ P (Ex. 697), and solving, the results tabulated below are found, showing the perimeters to five decimal places. The results of the last two computations show that the circumference of a circle whose diameter is 1 is approximately 3.1416; that is, the ratio of the diameter of a circle to its circumference is equal to the ratio of 1 to 3.1416 approximately. Ex. 699. The sides of an inscribed rectangle are 30cm and 40cm. is the area of the part of the circle without the rectangle? What Ex. 700. What is the area of a figure bounded by four semicircumferences described on the sides of a three foot square? Ex. 701. A square piece of land and a circular piece of land each contain one acre. Which perimeter is the greater, and how much? Ex. 702. The area of an inscribed equilateral triangle is one half the area of a regular hexagon inscribed in the same circle. Ex. 703. Of all triangles that have the same vertical angle and whose bases pass through a given point, the minimum is the one whose base is bisected at that point. Ex. 704. An arc of a circle whose radius is 6 ft. subtends a central angle of 20°; an equal arc of another circle subtends a central angle of 30°. What is the radius of the second circle? Ex. 705. Two tangents make with each other an angle of 60°, and the radius of the circle is 7 in. What are the lengths of the arcs between the points of contact ? Ex. 706. If the apothem of a regular hexagon is 10m, what is the area of the ring between the circumferences of its inscribed and circumscribed circles? Ex. 707. If a circle 18cm in diameter is divided into three equivalent parts by two concentric circumferences, what are their radii ? Ex. 708. The square upon the side of a regular inscribed pentagon is equivalent to the sum of the squares upon the radius of the circle and the side of a regular inscribed decagon. Ex. 709. The radius of a regular inscribed polygon is a mean proportional between its apothem and the radius of the similar circumscribed polygon. Ex. 710. If the radius of a regular inscribed octagon is r, prove that its side = r √2 √2, and its apothem r = √2+ √2. 2 Ex. 711. If the radius of a regular inscribed decagon is r, prove that its side (√5-1) and its apothem =√10+2 √5. = 4 Ex. 712. If the radius of a regular inscribed dodecagon is r, prove that its side =r√2 - √3, and its apothem r = √2 + √3. 2 Ex. 713. If the radius of a regular inscribed pentagon is r, prove that its √10 – 2 √5, and its apothem =√6 + 2√5. side = 4 Ex. 714. The square upon a side of an inscribed equilateral triangle is equivalent to three times the square upon the side of a regular inscribed hexagon. Ex. 715. The area of an inscribed square is 16sq m. Find the length of a side of a regular inscribed octagon. Ex. 716. If the radius of a circle is r, prove that a side of a regular 2r √3. circumscribed hexagon is 3 Ex. 717. The area of a regular inscribed dodecagon is equal to three times the square of the radius. MILNE'S GEOM.- -16 |