Or: The radius of the inscribed circle of a regular hecadecagon equals a side of the hecadecagon multiplied by the square root of two quantities, one of which is the square root of 2 added to 1 3/4, and the other is the square root of the sum of seven halves of the square root of 2 added to 5.
A = 16 HD x HF,
A = 16r b/2 = 8rb.
The value of r is shown in (148.); therefore we have -
Or: The area of a regular hecadecagon equals eight times the square of its side, multiplied by the square root of two quantities, one of which is the square root of 2 added to 1 3/4, and the other is the square root of the sum of seven halves of the square root of 2 added to 5.
439. - Polygons: Radius of Circumscribed and Inscribed Circles: Area. - In Arts. 433 to 438 the relation of the radii to a side in a trigon, tetragon, hexagon, octagon, dodeca-gon and hecadecagon have been shown by methods based upon geometrical proportions. This relation in polygons of seven, nine, ten, eleven, thirteen, fourteen and fifteen sides, cannot be so readily shown by geometry, but can be easily obtained by trigonometry - as also said relation of the parts in a regular polygon of any number of sides. The nature of trigonometrical tables is discussed in Arts. 473 and 474. So much as is required for the present purpose will here be stated.
Let ABC (Fig. 293) represent one of the triangles into which any polygon may be divided, in which B C = b = a side of the polygon; A C= R = the radius of the circumscribed circle; and A D = r = the radius of the inscribed circle.
Make E C equal unity; on C as a centre describe the arc EF; draw FH and E G perpendicular to B C, or parallel to A D; then for the uses of trigonometry E G is called the tangent of c, or of the angle A CB, and FH is the HC, and HC the cosine of the same angle.
These trigonometrical quantities for angles varying from zero up to ninety degrees have been computed and are to be found in trigonometrical tables.
Referring now to Fig. 293 we have -
Hc: Fc:: Dc: Ac,
cos. c: I :: b/2: R,
R = b
Ec: Eg:: Dc: Ad,
1: tan. c::b/2: r,
r = b/2 tan. c.
These two equations give the required radii of the circumscribed and inscribed circles. They may be stated thus:
The radius of the circumscribed circle of any regular polygon equals a side of the polygon divided by twice the cosine of the angle formed by a side of the polygon and a radius from one end of the side.
The radius of the inscribed circle of any regular polygon equals half of a side of the polygon multiplied by the tangent of the angle formed by a side of the polygon and a radius from one end of the side.
The area of a polygon equals the area of the triangle ABC (Fig. 293), (of which B C is one side of the polygon and A is the centre), multiplied by the number of sides in the polygon; or, if n be put to represent the number of the sides and A the area, then we have -
A = Bn,
in which B equals the area of the triangle. The area of A B C (Fig. 293) is equal to AD x B D, or -
For r substituting its value, as in equation (151.), we have -
B = b/2 tan. cb/2 = 1/4 b2 tan. c.
Therefore, by substitution -
A = - 1/4b2 n tan. c. (152.)
Or: The area of a regular polygon equals the square of a side of the polygon, multiplied by one fourth of the number of its sides, and by the tangent of the angle formed by a side of the polygon, and a radius from one end of the sides.
440. - Polygons: Their Angles. - Let a line be drawn from each angle of a regular polygon to its centre, then these lines form with each other angles at the centre, which taken together amount to four right angles, or to 360 degrees (Arts. 327, 335).
If this 360 degrees be divided by the number of the sides of the polygon, the quotient will equal the angle at the centre of the polygon, of each triangle formed by a side and two radii drawn from the ends of the side. For example: if ABC (Fig. 293) be one of the triangles referred to, having B C one of the sides of the polygon and the point A the centre of the polygon, then the angle B A C will be equal to 360 degrees divided by the number of the sides of the polygon. If the polygon has six sides, then the angle B A C will contain
360/6 = 60 degrees; or if there be 10 sides, then the angle at
A, the centre, will contain360/10= 36 degrees. The angle
BAD equals half the angle BAC, or, when n equals the number of sides, the angle BAC equals -
and the triangle B A D =BAC, equals -
Now the angles B A D + D B A equal one right angle (Art. 346), or 90 degrees. Hence the angle DBA =90° - BAD,or the angle c equals -
co=90o - 360o/2n. (153.)
For example, if n equal 6, or the polygon have six sides, then -
co = 90o - 360o = 90 - 30 = 60o
Therefore, the angle c, contained in equations (150.), (151.), and (152.), equals 90 degrees, less the quotient derived from a division of 360 by twice the number of sides to the polygon.
441. - Pentagon: Radius of the Circumscribed and Inscribed Circles: Area. - The rules for polygons developed in the two former articles will here be exemplified in their application to the case of a regular pentagon, or polygon of five sides.
To obtain the angle c° (153.), we have n = 5, and -
c° =90° -360 = 90- 36 = 54°.