The base of a triangular pyramid is a triangle with corners at $(6, 2)$ $(6 ,2 )$ , $(3, 8)$ $(3 ,8 )$ , and $(4, 4)$ $(4 ,4 )$ . If the pyramid has a height of $6$ $6$ , what is the pyramid's volume?

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Answer 1 · 2018-06-19T02:46:26

$color(purple)("Area of " Delta = A_t = 3 " sq units"$

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$"coordinates of three vertices are "$

$A(x,y) = 6,2), B(x,y) = (3,8), C(x,y) = (4,4)$

$a = sqrt ((3-4)^2 + (8-4)^2 = sqrt17 = 4.123$

$b = sqrt((4-6)^2 + (4-2)^2) = sqrt8 = 2.828$

$c = sqrt((6- 3)^2 + (2-8)^2) = sqrt45 = 6.708$

$"Semi-perimeter " = s = (a + b + c) / 2$

$s = (4.123 + 2.828 + 6.708) / 2 = 6.83$

$"Area of " Delta = A_t = sqrt (s (s-a) (s-b) (s - c)), " using the formula from the above table, knowing three sides"$

$A_t = sqrt (6.83 * (6.83 - 4.123) (6.83 - 2.828) (6.83 - 6.708))$

$color(purple)("Area of " Delta = A_t = 3 " sq units"$

The base of a triangular pyramid is a triangle with corners at (6 ,2 )(6,2)(6 ,2 ), (3 ,8 )(3,8)(3 ,8 ), and (4 ,4 )(4,4)(4 ,4 ). If the pyramid has a height of 6 66 , what is the pyramid's volume?