Fig. 290.

436.- - Octagon: Radius of Circumscribed and Inscribed Circles: Area. - Let CEDBF(Fig. 290) represent a quarter of a regular octagon, in which F is the centre, ED a side, and CE and D B each half a side, while CF and BF are radii of the inscribed circle, and EF and DF are radii of the circumscribed circle.

Let R represent the latter, and r the former; also let b represent ED, one of the sides, and n be put for A D, and for A E. Then we have -

AD+DB=CF

n + b/2 = r,

or - n = r-b/2

Since A D E is a right-angled triangle (Art. 416), we have -

AE2 + AD2 = ED2 n2+n2 = b2,

2n2 = b2,

n2 = b2/2 2

432 Perpendicular In Triangle Of Known Sides 361

Placing the value of n, equal to the value before found, we have -

432 Perpendicular In Triangle Of Known Sides 362

This coefficient may be reduced by multiplying the first fraction by

432 Perpendicular In Triangle Of Known Sides 363

thus -

432 Perpendicular In Triangle Of Known Sides 364

therefore -

(140.)

(140.)

Or: The radius of the inscribed circle of a regular octagon equals half a side of the octagon multiplied by the sum of unity plus the square root of 2. In regard to the radius of the circumscribed circle, by Art. 416 we have -

DF2 = BF2+DB2,

R2 = r2 + (b/2)2

In this expression substituting for r2, its value as above, we have -

432 Perpendicular In Triangle Of Known Sides 366

The square of the coefficient

432 Perpendicular In Triangle Of Known Sides 367

by Art. 412 equals

432 Perpendicular In Triangle Of Known Sides 368

then -

(141.)

(141.)

Or: The radius of the circumscribed circle of a regular octagon equals half a side of the octagon multiplied by the square root of the sum of twice the square root of 2 plus 4.

In regard to the area of the octagon, the figure shows that one eighth of it is contained in the triangle D E F.

The area of D E F, putting it equal to N, is -

N=EDxBF/2,

N=bxr/2,

432 Perpendicular In Triangle Of Known Sides 370432 Perpendicular In Triangle Of Known Sides 371

This is the area of one eighth of the octagon; the whole area, therefore, is -

432 Perpendicular In Triangle Of Known Sides 372(142.)

(142.)

Or: The area of a regular octagon equals twice the square of a side, multiplied by the sum of the square root of 2 added to unity. When a side of the enclosing square, or diameter of the inscribed circle, is given, a side of the octagon may be found; for from equation (140.), multiplying by two, we have -

432 Perpendicular In Triangle Of Known Sides 374

Dividing by

432 Perpendicular In Triangle Of Known Sides 375

+ 1, gives -

(143)

(143)

The numerator, 2 r, equals the diameter of the inscribed circle, or a side of the enclosing square; therefore:

The side of a regular octagon, equals a side of the enclosing square divided by the sum of the square root of 2 added to unity.

437. - Dodecagon: Radius of Circumscribed and Inscribed Circles: Area. - Let A B C (Fig. 291) be an equilateral triangle. Bisect A B in F; draw CFD; with radius A C describe the arc ABB. Join A and D, also D and B; bisect A D in E; with the radius E C describe the arc E G. Then A D and D B are sides of a regular dodecagon, or twelve-sided polygon; of which A C, D C, and B C are radii of the circumscribing circle, while E C is a radius of the inscribed circle.

Side And Area Of Dodecagon

The line A B is the side of a regular hexagon (Art. 435). Putting R equal to A C the radius of the circumscribing circle; r, = E C, the radius of the inscribed circle; b, = AD, a. side of the dodecagon, and n = D F. Then comparing the homologous triangles, A DF and A EC (the angle ADF equals the angle E A C, and the angles D F A and A E C are right angles); therefore, the two remaining angles DAF and A CE must be equal, and the two triangles homologous (Art. 345). Thus we have -

Side And Area Of Dodecagon 377

Fig. 291.

Df: Da :: Ae: A C,

n: b:: b/2: R,

R = b2/2 n

In Art. 435 it was shown that FC (Fig. 291), or GH of Fig. 289, the radius of the inscribed hexagon, equals

Side And Area Of Dodecagon 378

and in which its b == R; Fc =

Side And Area Of Dodecagon 379