10

For the radius of the circumscribed circle, we have (150.) -

R = b/2cos.c,

464. POLYGONS.

R = b/cos.54o

R = b I/2cos.54o

Using a table of logarithmic sines and tangents (Art. 427), we have -

Log. 2 =

0.3010300

Cos. 540 =

9.7692187

Their sum =

0.0702487

- subtracted from

Log. 1 =

0.0000000

0.85065 =

9.9297513

Therefore -

R = 0.85065 b.

Or: The radius of the circumscribed circle of a regular pentagon equals a side of the pentagon multiplied by the decimal 0.85065.

For the radius of the inscribed circle, we have (151.) -

r =b/2 tan. c,

r = b tan. 54o/2

For this we have -

Log. tan. 540 =

0.1387390

Log. 2 =

0.3010300

0.68819 =

9.8377090.

Therefore -

r = 0-68819 b.

Or: The radius of the inscribed circle of a regular pentagon equals a side of the pentagon multiplied, by the decimal 0.68819. For the area we have (152.) -

A = 1/4 b2n tan. c,

A = 1/4 x 5 tan. 54o b2,

A =5/4 tan. 54o b2 '

For this we have -

Log. 5. =

0.6989700

Log. tan 54o =

0.1387390

0.8377090

Log. 4 =

0.6020600

1.72048 =

0.2356490

Therefore -

A = 1.72048 b2.

Or: The area of a regular pentagon equals the square of its side multiplied by 1.72048.

1-42. - Polygons: Table of Constant Multipliers. - To obtain expressions for the radii of the circumscribed and inscribed circles, and for the area for polygons of 7, 9, 10, 11, 13, 14, and 15 sides, a process would be needed such precisely as that just shown in the last article for a pentagon, except in the value of n and c, which are the only factors which require change for each individual case.

No useful purpose, therefore, can be subserved by exhibiting the details of the process required for these several polygons. The values of the constants required for the radii and for the areas of these polygons have been computed, and the results, together with those for the polygons treated in former articles, gathered in the annexed Table of Regular Polygons.

Regular Polygons

Sides.

R/b=

r/b=

A/b2=

3.

Trigon...........................

.57735

.28868

.43301

4.

Tetragon..................

.70711

.50000

1.00000

5.

Pentagon.................

.85065

.68819

1.72048

6.

Hexagon

1.00000

.86603

1.72048

7.

Heptagon.................

1.15238

1.03826

3.63391

8.

Octagon.............

1.30656

1.20711

4.82843

9.

Nonagon.................

1.46190

1.37374

6.18182

10.

Decagon

1.61803

1.53884

7.69421

11.

Undecagon..............

1.77473

1.70284

9.36564

12.

Dodecagon..........

1.93185

1.86603

11.19615

13.

Tredecagon...........

2.08020

2.02858

13.18577

14.

Tetradecagon..............

2.24698

2.19064

15.33451

15.

Pentadecagon...........

2.40487

2.35231

17.64236

16.

Hecadecagon.............

2.56292

2.51367

20.10936

In this table R represents the radius of the circumscribed circle; r the radius of the inscribed circle; b one of the sides, and A the area of the polygon. By the aid of the constants of this table, R, the radius of the circumscribed circle of any of the polygons named, may be found when a side of the polygon is given. For this purpose, putting m for any constant of the table, we have -

R - bm. (154.)

As an example: let it be required to find R, for a pentagon having each side equal to 5 feet; then the above expression becomes -

R = 5 x 0.85065, R = 4.25325.

The radius will be 4 feet 3 inches and a small fraction. In like manner the radius of the inscribed circle will be -

r = bm; (155.)

and for a pentagon with sides of 5 feet, we have -

r = 5 x 0.68819, r = 3.44095.

Or, the radius of the inscribed circle will be 3 ft. 44/100 and a small fraction. Or, multiplying the decimal by 12, 3 ft. 5 in. 29/100 and a small fraction.

The area of any polygon of the table may be obtained by this expression -

A = b2m; (156.)

and, applying this to the pentagon as before, we have -

A = 52 x 1.72048, A = 43.012.

Or, the area of a pentagon having its sides equal to 5 feet, is 43 feet and 12/1000 of a foot.

By the constants of the table a side of any of its polygons may be found, when either of the radii, or the area, are known.

When R is known, we have -

b = R/m. (157.)

When r is known, we have -

b = r/m. (158.)

When the area is known, we have -

(159.)

(159.)