Plane trigonometry, by S.L. Loney.

REGULAR POLYGONS.

257

Again, the area
= n. OL. BL = n. OB cos LOB. OB sin LOB
7 r 7 n. 27r
= nR2 cos - sin - = R sin..............(3).
n n 2 n.
226. Ex. The length of each side of a regular dodecagon is 20 feet;
find (1) the radius of its inscribed circle, (2) the radius of its circumscribing
circle, and (3) its area.
The angle subtended by a side at the centre of the polygon
3600
1= 30~.
12
Hence we have 10 = r tan 15 = R sin 15~.
r=10 cot 15~
10
=2 -3 (Art. 101)
=10 (2+ V3)= 37-32... feet.
Also R = i s -10 x /^ 1 (Art. 106)
sin 150 x /3 -1
= 10. J/2 (J3 + 1) = 10 (J6 + V2)
=10 (2-4495... + 1 4142...)=38-637... feet.
Again, the area = 12 x r x 10 square feet
= 1200(2+ /3) =4478 46... square feet.
EXAMPLES. XXXIX.
1. Find, correct to '01 of an inch, the length of the perimeter of a
regular decagon which surrounds a circle of radius one foot.
2. Find to 3 places of decimals the length of the side of a regular
polygon of 12 sides which is circumscribed to a circle of unit radius.
3. Find the area of (1) a pentagon, (2) a hexagon, (3) an octagon,
(4) a decagon and (5) a dodecagon, each being a regular figure of side
1 foot.
4. Find the difference between the areas of a regular octagon and a
regular hexagon if the perimeter of each be 24 feet.
L. T. 17