Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Calculating Green Area to 3 significant figures.
Notice.
8 cm is the side length of the regular nonagon and also the regular bigger inscribed pentagon.
a = ⅑*180(9-2)
a = 140°
a is the single interior angle of the regular nonagon.
b = ⅕*180(5-2)
b = 108°
b is the single interior angle of the both inscribed regular pentagons.
c = 108-½(180-108)
c = 72°
tan72 = d/(0.5x)
d = 1.53884176859x cm.
d is the height of the big inscribed regular pentagon.
x is the side length of the big inscribed regular pentagon.
tan72 = e/4
e = 12.3107341487 cm.
e is the height of the biggest inscribed regular pentagon.
f = d+e
f = (1.53884176859x+12.3107341487) cm.
f is the height of the regular nonagon.
g = ½(360-2(140))
g = ½(80)
g = 40°
cos40 = h/8
h = 6.12835554495 cm.
j = 2h+8
j = 20.2567110899 cm.
k² = 4²+20.2567110899²-2*4*20.2567110899cos100
k = 21.318411571 cm.
(21.318411571/sin100) = (4/sinl)
l = 10.6483469999°
m = 100-l
m = 89.3516530001°
n² = 8²+21.318411571²-2*8*21.318411571cos89.3516530001
n = 22.6851272785 cm.
n = f, the height of the regular nonagon.
Calculating x, equating n and f.
1.53884176859x+12.3107341487 = 22.6851272785
1.53884176859x = 22.6851272785-12.3107341487
1.53884176859x = 10.3743931298
x = 10.3743931298/1.53884176859
x = 6.74168932866 cm.
Again, x is the side length of the inscribed big regular pentagon.
o = ½(a-b)
o = ½(140-108)
o = ½(32)
o = 16°
Therefore, the big green area is;
0.5*8*6.74168932866sin16
= 7.43304568117 cm²
Calculating the bigger green area.
p = 180-108
p = 72°
(8/sin72) = (q/sin54)
q = 6.80520646682 cm.
r²+4² = 6.80520646682²
r = 5.50552768189 cm.
(6.74168932866/sin72) = (s/sin54)
s = 5.73482347708 cm.
t = n-r-s
t = 22.6851272785-5.50552768189-5.73482347708
t = 11.4447761195 cm.
t is a side length of the bigger green area.
u² = 8²+5.73482347708²-2*8*5.73482347708cos(0.5*140)
u = 8.09354050784 cm.
(8.09354050784/sin70) = (8/sinv)
v = 68.2535006534°
w = 180-70-v
w = 180-70-68.2535006534
w = 41.7464993466°
x' = (140-w)+0.5(108)
x' = (140-41.7464993466)+0.5(108)
x' = 152.253500653°
y² = 8.09354050784²+6.80520646682²-2*6.80520646682*8.09354050784cos152.253500653
y = 14.4674269963 cm.
y is a side length of the bigger green area.
(14.4674269963/sin152.253500653) = (6.80520646682/sinx)
z = 12.6497710931°
z' = 180-v-z
z' = 180-68.2535006534-12.6497710931
z' = 99.0967282535°
It implies;
Bigger green area is;
0.5*t*ysinz'
= 0.5*11.4447761195*14.4674269963sin99.0967282535
= 81.7469903673 cm²
Area Green Total is;
7.43304568117+81.7469903673
= 89.1800360485 cm²
Therefore, area green to 3 significant figures is;
≈ 89.2 cm²
