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?

1 Answer
Nov 17, 2017

36 units cubed

Explanation:

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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)