Question

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

`36` units cubed

Explanation:

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)`