The base of a triangular pyramid is a triangle with corners at #(2 ,1 )#, #(5 ,2 )#, and #(8 ,7 )#. If the pyramid has a height of #18 #, what is the pyramid's volume?

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Answer 1 · 2017-11-17T13:56:13

#36# units cubed

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Volume of a triangular pyramid is:

#V=1/3Ah#

Where #A =# area of base, and #h=# height.

Dimensions of triangle:

Let the angles be at:

#A=( 2 , 1 )#

#B= ( 5 , 2 )#

#C= ( 8 , 7 )#

Using the distance formula:

Length #AB#

#AB=sqrt((2-5)^+(1-2)^2)=sqrt(10)#

#BC=sqrt((5-8)^2+(2-7)^2)=sqrt(34)#

#AC=sqrt((8-2)^2+(7-1)^2)=sqrt(72)=6sqrt(2)#

Finding #sin(A)# of angle #A# using the cosine rule:

#cos(A)=(b^2+c^2-a^2)/(2bc)#

#cos(A)=((6sqrt(2))^2+(sqrt(10))^2-(sqrt(34))^2)/(2(6sqrt(2))(sqrt(10)))#

#->=(72+10-34)/(24sqrt(5))=48/(24sqrt(5))=2/sqrt(5)#

#sin(A)=sin(cos^-1(2/sqrt(5)))=sqrt(5)/5#

Area of triangle from diagram:

#1/2(sqrt(10))sin(A)b#

#Area=1/2(sqrt(10))(sqrt(5)/5)(6sqrt(2))=60/10=6#

Area of pyramid:

#1/3(6)(18)=36color(white)(88)#