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 Trespectively.

and substituting for AE, AB, and EF their values,

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, Q = 4, and P=2 √2 for the octagon.

Substituting these values in the formulæ,

T=2Q×P
+P

S=√PXT

(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 V2 √2, and its apothem

=

√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="√2 + √3.

2

Ex. 713. If the radius of a regular inscribed pentagon is r, prove that its side = √10-2√5, and its apothem =√6 +2√5.

2

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.

side of a regular inscribed octagon.

Find the length of a

Ex. 716. If the radius of a circle is r, prove that a side of a regular

Ex. 717. The area of a regular inscribed dodecagon is equal to three times the square of the radius.

Ex. 718. Find the side of a regular hexagon circumscribed about a circle whose diameter is 1.

Ex. 719. The apothem of an inscribed regular hexagon is equal to one half the side of the inscribed equilateral triangle.

Ex. 720. The area of a ring bounded by two concentric circumferences is equal to the area of a circle whose diameter is a chord of the outer circumference and is tangent to the inner circumference.

Ex. 721. If the radius of a circle is r, find the area of a segment whose chord is one side of a regular inscribed hexagon.

Ex. 722. Three equal circles with a radius of 12 ft. are drawn tangent to each other. What is the area between them?

Ex. 723. The area of an inscribed regular hexagon is equal to three fourths that of a regular hexagon circumscribed about the same circle.

Ex. 724. The altitude of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle whose diameter is the base of the first triangle.

Ex. 725. If the radius of a circle is r and the side of a regular inscribed 2 ar polygon is a, prove that the side of a similar circumscribed polygon is

√4r2-a2

Ex. 726. If the alternate vertices of a regular hexagon are joined by straight lines, another regular hexagon is formed which is one third as large as the original hexagon.

Ex. 727. The diagonals of a regular pentagon divide each other in extreme and mean ratio.

PROBLEMS OF CONSTRUCTION

= √ab.

Ex. 728. Construct x, if x =

Ex. 729. Inscribe a circle in a given sector.

Ex. 730. In a given circle describe three equal circles tangent to each other and to the given circle.

Ex. 731. Divide a circle into two segments such that an angle inscribed in one shall be three times an angle inscribed in the other.

Ex. 732. Construct a circumference equal to the sum of two given circumferences.

Ex. 733. Inscribe a square in a given quadrant.

Ex. 734. Inscribe a square in a given segment of a circle.

Ex. 735. In an isosceles triangle describe three circles touching each other

and each touching two of the three sides of the triangle.

Ex. 736. Construct a circle equivalent to twice a given circle.

Ex. 737. Construct a circle equivalent to three times a given circle.

SOLID GEOMETRY

BOOK VII

PLANES AND SOLID ANGLES

427. A plane is a surface such that a straight line joining any two of its points lies wholly in the surface. § 14.

A plane is considered to be indefinite in extent, but in a diagram it is usually represented by a quadrilateral segment.

428. The student will be aided in obtaining correct concepts of the truths presented in the geometry of planes by using pieces of cardboard or paper to represent planes, and drawing such lines upon them as are required. Pins may be used to represent the lines which are perpendicular or oblique to the planes.

429. 1. By using cardboard to represent a plane and the point of a pin or pencil to represent a point in space, discover in how many directions the plane may be passed through the point.

2. By using a card as before and the points of a pair of dividers to represent two fixed points in space, discover whether the number of directions that the plane may take is greater or less than when it was passed through one fixed point.

3. Suppose a plane is passed through three fixed points not in the same straight line, how many directions may it take? How many points, then, determine the position of a plane?

4. Since two of the points must be in a straight line, what else besides three points determine the position of a plane?

5. Since a straight line through the other point may intersect the straight line joining the two points, what else will determine the position of a plane?