Hemisphere radius = circle diameter = square diagonal = square area = 2
[ADE] / [ABCD] = 1/6
The circumference of the cylinder's base is 6 units.
Radius = 1/2(6/pi)
Volume = r2pi*h
h = (27/2) / r2pi
sin(∠BAD) = h / 6
Nice work, heureka, as always!!!
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QS = TU = 2
QT = QR = 8
QW = 5
RV = 4
TW = sqrt(QT2 - QW2)
RT = sqrt(TW2 + RW2)
RW / RT = RV / PR
PQ = PR = (RT * RV) / RW
Regular pentagon ABCDE is inscribed in a unit circle. Compute AB*AC.
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I find this very interesting:
If the length of a side of a regular pentagon is 1 unit, then the length of AC = 1.618033989 Golden ratio
Hint::: Triangle ABC is an equilateral triangle!
Side = 2(27√3 / √3)
Height AD = sin60 * 10
CD = cos60 * 10
BD = sqrt(AB2 - AD2)
a = CD + BD
Height BX = sinA * 4 = 2
[ABC] = 1/2(2 * 5) = 5
Height EY = sinD * 4 = 3.2
[DEF] = 1/2(3.2 * 5) = 8
[ADE] ≈ 21.108
[BDP] ≈ 22.154
Golden ratio
+1693 
