Home > Geometry > Volume of Solids

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?

Answer 1

Claire Akin

#36# units cubed

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)#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Carson Arnold

The volume of the triangular pyramid is 144 cubic units.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.