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# A regular hexagon with sides of length 6 has an isosceles triangle att

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Math Expert
Joined: 02 Sep 2009
Posts: 55230
A regular hexagon with sides of length 6 has an isosceles triangle att  [#permalink]

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25 Apr 2019, 03:22
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Difficulty:

55% (hard)

Question Stats:

47% (02:49) correct 53% (02:48) wrong based on 17 sessions

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A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

(A) 18
(B) 162
(C) $$36\sqrt{21}$$
(D) $$18\sqrt{138}$$
(E) $$54\sqrt{21}$$

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Senior Manager
Joined: 25 Feb 2019
Posts: 280
Re: A regular hexagon with sides of length 6 has an isosceles triangle att  [#permalink]

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25 Apr 2019, 18:56
1
IMO C .

volume of pyramid= 1/3(area of base) *height

here area of base = area of regular hexagon = 6*(√3/4)(6^2)

for height of pyramid

drwa a perpemdicular fro. vertex of pyramid to its base and then join the toucg pint

to on of hexagone vertex ,

we get one rightangle triangle

where perp = 8 height = h ( assume) and third side = 6

so we have h =√(64-36) = 2√7

now out the values of area and height in above mentioned formula

we get 36√21

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Senior Manager
Joined: 19 Oct 2018
Posts: 261
Location: India
Re: A regular hexagon with sides of length 6 has an isosceles triangle att  [#permalink]

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26 Apr 2019, 00:44
Volume of pyramid= 1/3 *area of base*height
area of base(hexagon)= {3*(3^0.5)*side^2}/2=54*(3^0.5), side=6
h=(l^2- d^2)^1/2
l= slant edge of pyramid= length of equal sides of isosceles triangles=8
s= distance of any vertice of hexagon from midpoint of hexagon= 6
h=(8^2 - 6^2)^1/2= 28^1/2= 2(7^1/2)
volume of pyramid=1/3 * 54*(3^0.5) * 2*(7^0.5)=36(21^0.5)
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Joined: 12 Sep 2017
Posts: 246
Re: A regular hexagon with sides of length 6 has an isosceles triangle att  [#permalink]

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10 May 2019, 16:50
m1033512 wrote:
IMO C .

volume of pyramid= 1/3(area of base) *height

here area of base = area of regular hexagon = 6*(√3/4)(6^2)

for height of pyramid

drwa a perpemdicular fro. vertex of pyramid to its base and then join the toucg pint

to on of hexagone vertex ,

we get one rightangle triangle

where perp = 8 height = h ( assume) and third side = 6

so we have h =√(64-36) = 2√7

now out the values of area and height in above mentioned formula

we get 36√21

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Hello m1033512 !!!

Kind regards!
Re: A regular hexagon with sides of length 6 has an isosceles triangle att   [#permalink] 10 May 2019, 16:50
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