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Take radius CF = OD and describe the circle CF, then, since FG is included between radii AC and CB,

angle C : angle 0 :: FG : DE;


or, since CF = DO,

FG DE angle C : angle 0 ::

But, by reason of similar arcs,


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Cor. The

may be taken as the measure of the angle. radius

The unit of this measure is the arc equal in length to the radius. This measure of angles is called Circular measure; or, better, Radial

To express the unit in degrees, minutes, and seconds, we have semi-circumference =aR = 180°.

180° 180° .. R=

= 57° 17' 44".8 = 3437'.8 = 206254".8. 3.1416



These values of the radius in degrees, minutes, and seconds, are of the greatest importance in numerical problems connecting the measure of the circle and of angles.



1. An inscribed equiangular polygon is regular if the number of its sides be odd.

2. A circumscribed equilateral polygon is regular if the number of sides be odd.

3. The diagonals of a regular hexagon divide each other in the ratio of two to one.

4. The regular inscribed hexagon is double the equilateral triangle inscribed in the same circle, and one-half of the circumscribed equilateral triangle.

5. The regular inscribed hexagon is three-fourths of the regular hexagon circumscribed about the same circle.

6. The side of the circumscribed equilateral triangle is double the side of the inscribed equilateral triangle, and the altitude is three times the radius of the circle.

7. The area of the regular inscribed dodecagon is equal to three times the square of the radius.

8. The square of the side of the regular inscribed decagon, together with the square of the radius, is equal to the square of the side of the regular inscribed pentagon.

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

10. If we join the first, fourth, seventh, etc., vertices of a regular inscribed decagon, each one of the joining chords is equal to the radius of the circle plus the side of the decagon. And, applying this chord to the circumference, after going around three times the extremity will fall at the point of starting. And thus a regular reentrant decagon will be formed (called a star decagon).

11. If the first, fifth, ninth, etc., vertices be joined, we form the star pentagon, or we effect the same by joining the alternate vertices of the above star decagon.

12. If a circumference be divided into five equal parts, and the points of division, A, B, C, D, E, be joined by the lines AC, CE, EB, BD, DA, these lines will form by their intersection a regular pentagon.

13. If, from any point within a regular polygon of n sides, we let fall perpendiculars on the sides, the sum of these perpendiculars will be equal to n times the apothem of the polygon.

14. If, on the side of a square, we take distances from the vertices equal to one-half of the diagonal, and join the points thus taken on adjacent sides, we form a regular octagon.

15. If two circles cut each other at right angles (that is, if their tangents at the point of intersection be perpendicular to each other), and if the distance between the centres be double one of the radii, then the common chord, is the side of a regular hexagon in the greater circle, and of an equilateral triangle in the other.

16. If we describe two equal semi-circumferences on the diameter of a given semicircle, and then inscribe a circle in the space between these three semicircles, touching the three semi-circumferences, the diameter of this circle will be one-third of the diameter of the first circle.

17. If, on the sides of a triangle, ABC, right angled at C, we describe three semicircles, AMCNB, AOC, and BPC (these last two exterior to the triangle), then the curvilinear spaces, cut off by the semi-circumference on the hypothenuse from the other two semicircles, are together equal to the area of the triangle (these spaces are called the Lunules of Hippocrates).

18. Show, by the consideration of the perimeters of the regular inscribed hexagon and circumscribed square, that the value of a is comprised between the numbers 3 and 4.

19. Show that the semi-circumference of the circle is nearly equal to the sum of the side of the inscribed equilateral triangle and the side of the inscribed square.

20. If we divide the diameter, AB, of a circle into two parts, AC, CB, and describe semi-circumferences on AC and CB, on different sides of AB: ist, the curve line composed of these two semi-circumferences divides the circle into two parts proportional to AC and CB ; and 2d, this line is also equal to the semi-circumference on · AB.

21. If we divide the diameter, AB, of a circle into any number of equal parts, as, for example, five, so that the points of division fall at C, D, E, and F, and if upon AC, AD, AE, and AF, we describe semi-circumferences on one side of AB, and semi-circumferences on BC, BD, BE, and BF, all falling on the other side of AB, then the curve lines made up of these semi-circumferences, two and two respectively, shall divide the circle into five equal parts.

22. A circular ring (that is, the space included between two concentric circles) is equal to the circle which has for its diameter the chord of the greater circle which is tangent to the smaller.

23. If we describe a semi-circumference on CA, the radius of a given circle whose centre is C, and, dividing this radius into any number of equal parts, for example, into four, erect perpendiculars at the points of division, meeting the semi-circumference on AC in the points M, N, and 0 : then the circles described with the centre C and radii AM, AN, and AO, respectively, will divide the circle of radius CA into four equal parts.

24. If we make a circumference roll along a fixed circumference of double radius, and within it, then, a point on the rolling circumference will describe a diameter of the fixed circle.


1. In an equilateral triangle inscribe three equal circles touching one another, and each touching two sides of the triangle.

2. In a given circle inscribe three equal circles touching one another and the given circle.

3. In a given square inscribe four equal circles tangent each to two of the others and to one side of the square.

4. In a given circle inscribe four equal circles, tangent each to two of the others and to the given circle.

5. Compute the side of a regular decagon, the radius of the circle being R; also the side of the regular pentagon.

6. Determine by a single and the same construction the side of the regular decagon, and the side of the regular pentagon inscribed in a given circle.

7. Compute the area of the regular inscribed hexagon, octagon, dodecagon, and equilateral triangle, the radius of the circle being 2.4 meters. Make the computation also in feet (the meter being 39.37 inches nearly).

8. Three equilateral triangles whose sides are respectively 3 feet, 5

feet, and 2 feet, being given, find the equilateral triangle equal to their sum. 9.

Radius of circle being 3.20 meters, compute to within .001 meter the area of inscribed and circumscribed equilateral triangle, and area of regular inscribed and circumscribed hexagon (compute the same in feet).

10. Area of regular dodecagon being 3888 square meters, find the area of regular decagon inscribed in the same circle.

11. Construct a circle equivalent to the sector of a given circle whose arc is 32° 24'.

12. The fore wheels of a carriage have a radius of 24 centimeters, the hind wheels a radius of 40 centimeters ; how many revolutions does each make in five kilometers.

13. The area of a regular hexagon being 6400 square feet, find the area of the circle circumscribed about it.

14. Find the area of a segment whose arc is 60°, the radius of the circle being 512 meters.

15. Find the area of a circle in which the area of the sector whose arc is 36° 40', is 1200 square meters.

16. A circle, a square, and an equilateral triangle, all have the same perimeter, equal to one meter. Compare their areas.

17. The radius of the circumscribed circle being unity, compute the side and apothem of each of the following regular polygons :

First, Equilateral triangle ; Second, Square; Third, Octagon ; Fourth, Pentagon ; Fifth, Decagon ; Sixth, Dodecagon ; Seventh, Polygon of 20 sides.

18. Three equal circumferences, with the radius 6 inches, touch each other. Compute the area inclosed between them.

19. Four equal circles are inscribed in a square, each touching two of the others, the side of the square being four inches. Compute that part of the surface of the square which is exterior to the circles.

20. If an arc of 45° on one circumference is equal to an arc of 60° on another circle, what is the ratio of the areas of the circles.

21. Find the number of degrees, minutes, and seconds in the angle 1 (the unit of measure being the angle corresponding to the arc equal to the radius).

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