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428. Cor. If R=1, and the side of a polygon of n sides equals sn, then
8:27= V2 – V4–5
Ex. 961. The side of an inscribed square is equal to RV2. Compute the side of the regular octagon.
PROPOSITION XIX. PROBLEM 429. To compute the ratio of the circumference of a circle to the diameter. Let
R=1, then 86 = 1, and according to (428), we have
812 =V2 – V4–12 = 0.51764 6.21166
S768 = V2 – V4 –(0.01636)2 = 0.00818 6.28317
Taking the last perimeter as the approximate value of the circumference,
C_6.28317 = 3.14159. = 2R 2
430. REMARK. — Most computations referring to the circle involve the use of the two formulæ C= 2 TR and S = aR2. If R is not given, compute R first. (421.)
Ex. 962. Find the circumference of a circle whose radius is 5. .
Ex. 965. Find the radius of a circle whose circumference equals m inches.
Ex. 966. Find the circumference of a circle whose area equals S.
Ex. 968. Find the radius of a circle equivalent to a square whose side equals 6.
Ex. 969. The circumference of a circle equals 10. Find the circumference of a circle having twice the area of the given circle.
Ex. 970. Two concentric circles have their circumferences equal to 30 and 40 respectively. Find the area bounded by the two circumferences.
Ex. 971. Find a semicircle equivalent to an equilateral triangle whose side equals 5.
Ex. 972. Find the area of a sector whose radius equals 5, and whose central angle equals 40°.
Ex. 973. A square is inscribed in a.circle of radius 10. Find the area of a segment bounded by a side of the square.
Ex. 974. From a point without a circle, two tangents are drawn to a circle whose radius equals 10. Find the area bounded by the two tangents and the circumference, if the tangents include an angle of 120°.
Ex. 975. Find the area bounded by three arcs of 60° and radius 5, if the concave sides of the arcs are turned toward the area.
Ex. 976. Find the area bounded by three arcs of 60° and radius 5, if the convex sides of the arcs are turned toward the area.
Ex. 977. A circle has an area of 60 sq. in. Find the length of an arc of 40°.
Ex. 978. Find the central angle of a sector whose perimeter is equal to the circumference.
Ex. 979. Find the central angle of a sector whose area is equal to the square of the radius.
Ex. 980. If in a circle two chords intersect at right angles, and circles are constructed on the segments of the chords as diameters, the area of the given circle is equivalent to the sum of the areas of the four circles.
Ex. 981. The side of the regular circumscribed triangle is equal to twice the side of the regular inscribed triangle.
Ex. 982. The square of a side of an inscribed equilateral triangle is equal to three times the square of the inscribed regular hexagon.
Ex. 983. If the radius of a regular octagon be equal to r, prove that its side equals rV2 – V2.
Ex. 984. If the radius of a regular decagon be equal to r, prove that its side equals( V5 – 1).
Ex. 985. If the radius of a regular pentagon be equal to r, prove that its side equals V10 – 2V5.
Ex. 986. If the radius of a regular dodecagon be equal to r, prove that its side equals rV2 – 13.
Ex. 987. If in a circle a regular desagon and a regular pentagon be inscribed, the side of the decagon increased by the radius is equal to twice the apothem of the pentagon.
* Ex. 988. The square of the side of a regular decagon increased by the square of the radius is equal to the square of the side of a regular pentagon, having the same radius.
APPENDIX TO PLANE GEOMETRY
ALGEBRAIC SOLUTIONS OF GEOMETRICAL PROP
OSITIONS. MAXIMA AND MINIMA
CONSTRUCTION OF ALGEBRAIC EXPRESSIONS
431. Note. - In the following propositions, a, b, c, d, etc., denote given lines, while x, y, z, etc., denote required lines.
432. (1) Construct x = a + b. (2) Construct
X = a – b. (3) If m denotes a given number. Construct
X = m · a.
Hint. — * is the fourth proportional to c, a, and b.
(7) Construct x = Vab.
(Three methods : 307 and 314.) (8) Construct x = Va’ + bo. (9) Construct x = Va’ — 6?. (10) Construct x = av2. (11) If m denotes a known number.
Construct x = avm. Hint.
x = Va(am).
Ex. 991. Construct x =
V3 by means of an equilateral triangle.
433. Complex algebraical expressions are constructed by means of the eleven constructions of (432). It is quite often necessary to transform the algebraical expressions, in order to make them special cases of (432). Different algebraical transformations lead to different solutions.
x = V3 ab,
X = V(3 a)b, i.e. x is the mean proportional between 3 a and b, or
2 = Vab V3, i.e. find Vab by means of (432, 7), and Vab V3 by means of (Ex. 991). Ex. 993.
m_a? — 62 _ (a + b)(a - b)
sc = Va? – ab. Hint.
x = Vasa – b). Ex. 995. x = Vaz – bc = Va? – (Vbc)2, i.e. find the mean proportional between b and c, and construct a right triangle, having one arm equal to the mean proportional, and the hypotenuse equal to a.