The base of a triangular pyramid is a triangle with corners at $(5, 1)$ $(5 ,1 )$ , $(2, 3)$ $(2 ,3 )$ , and $(3, 4)$ $(3 ,4 )$ . If the pyramid has a height of $4$ $4$ , what is the pyramid's volume?

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Answer 1 · 2017-06-30T06:11:48

The volume is $= 3.33 u^{3}$ $=3.33u^3$

The area of the base triangle is calculated with the determinant

Therefore, the area of the base is

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

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

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

$=1/2(5(3-4)-1(2-3)+1(8-9))$

$=1/2(-5+1-1)$

$=1/2|-5|=5/2$

The volume of the pyramid is

$=1/3*h*A=1/3*4*5/2=10/3=3.33u^3$

The base of a triangular pyramid is a triangle with corners at (5 ,1 )(5,1)(5 ,1 ), (2 ,3 )(2,3)(2 ,3 ), and (3 ,4 )(3,4)(3 ,4 ). If the pyramid has a height of 4 44 , what is the pyramid's volume?