Find the diameter of a circle

13. The diameter of a circle is 12 feet. having (1) twice its circumference, (2) twice its area.

14. A circular pond is surrounded by a gravel walk, such that the area of the walk equals the area of the pond. What is the ratio of the diameter of the pond to the width of the walk?

15. A bicycle wheel is 28 inches in diameter. will it make in going 3 miles ?

How many revolutions

16. Three circles are concentric and are such that the area of the first equals the area between the first and the second, and also between the second and third. The radius of the smallest is 10 inches. What are the radii of the other two?

17. A circle is inscribed in a given square. of the square lies outside of the circle? enlarging the square and the inscribed circle?

What fraction of the area Is the fraction changed by

18. A square is inscribed in a given circle. What fraction of the area of the circle lies outside of the square?

19. Show that the altitude of an equilateral triangle is to the radius of the circumscribed circle in the ratio of 3 to 2.

20. Prove that the area of a circular ring is equal to the area of a circle whose diameter equals a chord of the outer boundary which is tangent to the inner.

21. Prove that any equilateral polygon circumscribing a circle is regular.

22. Each side of an inscribed equilateral triangle is parallel to the tangent at the opposite vertex.

23. Describe a circle whose area is equal to the sum of the areas of two given circles.

24. Describe a circle whose circumference is equal to the sum of the circumferences of two given circles.

25. Prove that the radius of an inscribed regular polygon is a mean proportional between its apothem and the radius of a similar circumscribed polygon.

26. If squares are described outwardly on the sides of a regular hexagon, prove that the outer vertices of the squares are the vertices of a regular dodecagon.

27. Find in degrees the angle at the centre of a circle of radius 2 feet, which is subtended by an arc whose length is 18 inches.

1. DEFINITIONS.

SUMMARY OF CHAPTER V

(1) Regular Polygon-one which is both equilateral and equiangular. § 335.

(2) Centre of a Regular Polygon - the common centre of the inscribed and circumscribed circles. § 344.

(3) Radius of a Regular Polygon — the radius of the circumscribed circle, i.e. the line from the centre to a vertex. § 345.

(4) Apothem of a Regular Polygon — the radius of the inscribed circle, i.e. the perpendicular from the centre on a side. § 346. (5) Circumference of a Circle — the limit of the perimeter of a regular inscribed polygon as the number of its sides is indefinitely increased. § 358.

Or, the limit of the perimeter of a regular circumscribed polygon as the number of its sides is indefinitely increased. § 360. (6) Length of an Arc of a Circle - the limit of the sum of chords in the arc as the number of such chords is indefinitely increased. § 358.

(7) Area of a Circle - the surface enclosed by the circle, equal to the limit of the area of a regular inscribed polygon as the number of its sides is indefinitely increased. § 359.

(9) Similar Segments those which have similar arcs. § 368.
(10) Similar Sectors — those which have similar arcs. § 369.
(11) Principal Diagonals of a Polygon ·

- the diagonals joining pairs

of opposite vertices in a polygon of an even number of sides. § 380.

(12) Pentagon, Hexagon, Octagon, Decagon, Dodecagon. See § 336.

2. POSTULATES.

(1) A circle may be divided into any given number of equal parts. (Postulate 8.) § 337.

3. PROBLEMS.

(1) To inscribe a square in a given circle. § 373.

(2) To inscribe a regular hexagon in a given circle. § 376.
(3) To inscribe an equilateral triangle in a given circle. § 378.
(4) To inscribe a regular decagon in a given circle. § 381.
(5) To inscribe a regular pentagon in a given circle. § 382.

(6) To inscribe a regular polygon of fifteen sides in a given circle. § 383.

(7) To find the length of a side of an inscribed regular polygon of 2 n
sides, having given the length of the side of the inscribed poly-
gon of n sides. § 384.

(8) To find approximately the ratio of the circumference to the
diameter of a circle, i.e. the value of π.
§ 385.

The last two are numerical rather than geometrical problems.

4. THEOREMS ON INSCRIBED AND CIRCUMSCRIBED REGULAR POLYGONS. (1) If a circle is divided into any number of equal arcs, (1) the chords joining the points of division taken in order form a regular inscribed polygon, (2) the tangents to the circle at the points of division taken in order form a regular circumscribed polygon. § 339.

(2) An equilateral polygon inscribed in a circle is also equiangular, and hence regular. § 342.

(3) A circle can be circumscribed about any regular polygon, and another circle can be inscribed in it. § 343.

(4) If the number of sides of a regular inscribed polygon be doubled, the perimeter will be increased, but if the number of sides of a regular circumscribed polygon be doubled, the perimeter will be diminished. § 354.

(5) The area of a regular inscribed polygon is increased, and the area of a regular circumscribed polygon is diminished, when the number of sides is doubled. § 355.

(6) The side of an inscribed square is equal to r√2. § 374.

(7) All squares inscribed in the same circle are identically equal. $ 375.

(8) The side of a regular inscribed hexagon is equal to a radius of the circle. § 377.

(9) The side of an inscribed equilateral triangle is equal to r√3, and its distance from the centre is r. § 379.

5. THEOREMS ON THE PROPERTIES OF REGULAR POLYGONS.

(1) Any radius of a regular polygon bisects the angle at the vertex. § 347.

(2) The angle formed by two consecutive radii of a regular polygon equals four right angles divided by the number of sides of the polygon. § 348.

(3) Any two regular polygons of the same number of sides are

similar. § 349.

(4) Homologous sides in two regular polygons of the same number of sides are in the same ratio as the radii of the circumscribed circles, or as the radii of the inscribed circles; that is, as the radii of the polygons or as the apothems of the polygons. § 350. (5) The perimeters of two regular polygons of the same number of sides are in the same ratio as their radii, or as their apothems. § 351.

(6) The areas of two regular polygons of the same number of sides are in the same ratio as the squares of their radii, or as the squares of their apothems. § 352.

(7) The area of a regular polygon is equal to half the product of its perimeter and its apothem. § 353.

6. THEOREMS ON THE CIRCUMFERENCES AND AREAS OF CIRCLES. (1) The circumference of a circle is greater than the perimeter of any regular inscribed polygon, and less than the perimeter of any regular circumscribed polygon. § 361.

(2) The ratio of the circumference of a circle to its diameter is the
same for all circles. C=πd=2 π. § 362.

(3) The area of a circle is equal to one-half the product of its circum-
ference and radius. § 364. A = { Сr = πr2 = πd2. § 365.
(4) The areas of two circles are in the same ratio as the squares of
their radii, or as the squares of their diameters. § 366.

7. THEOREMS ON SECTORS OF A CIRCLE.

(1) The area of a sector of a circle bears the same ratio to the area of the circle as the length of its arc bears to the whole circumference. § 370.

(2) The area of a sector of a circle equals one-half the product of its arc and radius. S=ar. § 371.

(3) The arcs of similar sectors are in the same ratio as the radii, and the areas of similar sectors are in the same ratio as the squares of the radii. § 372.

PART II-SOLID GEOMETRY

CHAPTER VI

LINES AND PLANES IN SPACE

SECTION I

INTERSECTING PLANES-PARALLELS AND PERPENDICULARS

386. We come now to the study of geometrical figures whose points and lines do not all lie in the same plane. Such a figure is called a solid figure, or a figure of three dimensions.

A solid figure is spread out or extended in three ways.

A plane figure is two-dimensional since it is spread out in but two ways.

A straight line is one-dimensional since it has only length.

387. It should be remembered that in geometry we are concerned only with forms and relations, not at all with matter; and so it is necessary to distinguish between a three-dimensional figure, or a so-called solid figure, and a physically solid body.

A solid figure is a combination of points, lines, and surfaces. It contains no matter.

Since in Solid Geometry the diagrams are intended to represent figures which do not lie wholly in one plane, they are drawn as we say in perspective, to give an idea of how the figure would look at a distance. Straight lines and planes are supposed to extend indefinitely in any of their directions, though they must be represented in a diagram by limited portions of a line or a plane. For the most part lines are dotted when they are supposed to be seen through a surface which forms a part of the figure.