The base of a triangular pyramid is a triangle with corners at $(3, 5)$ $(3 ,5 )$ , $(4, 2)$ $(4 ,2 )$ , and $(2, 6)$ $(2 ,6 )$ . If the pyramid has a height of $7$ $7$ , what is the pyramid's volume?

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Answer 1 · 2017-06-29T09:50:54

The volume is $= 2.33 u^{3}$ $=2.33u^3$

The area of the base triangle is calculated with the determinant

Therefore, the area of the base is

$A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|$

$A=1/2|(3,5,1),(4,2,1),(2,6,1)|$

$=1/2(3*|(2,1),(6,1)|-5*|(4,1),(2,1)|+1*|(4,2),(2,6)|)$

$=1/2(3(2-6)-5(4-2)+1(24-4))$

$=1/2(-12-10+20)$

$=1/2|-2|=1$

The volume of the pyramid is

$=1/3*h*A=1/3*7*1=7/3=2.33u^3$

The base of a triangular pyramid is a triangle with corners at (3 ,5 )(3,5)(3 ,5 ), (4 ,2 )(4,2)(4 ,2 ), and (2 ,6 )(2,6)(2 ,6 ). If the pyramid has a height of 7 77 , what is the pyramid's volume?