The base of a triangular pyramid is a triangle with corners at $(7 ,6 )$ , $(4 ,1 )$ , and $(3 ,2 )$ . If the pyramid has a height of $6$ , what is the pyramid's volume?

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Answer 1 · 2018-01-27T09:06:03

$color(brown)(V = 32$ cubic units

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Area of triangle base $B = sqrt(s (s-a) (s-b) (s-c))$ where

a, b c are the lengths of three sides and s the semi perimeter of the triangle $s = (a + b + c) /2$

To find the three sides by applying the distance formula,

$d = sqrt((x2-x1)^2 + (y2-y1)^2)$

$a = sqrt((3-4)^2 + (2-1)^2) = sqrt2 ~~ color(red)(1.4142$

$b = sqrt((3-7)^2 + (2-6)^2) = sqrt32 ~~ color(red)(5.6568$

$c = sqrt((4-7)^2 + (1-6)^2) = sqrt34 ~~ color(red)(5.831$

Semi perimeter of triangle

$s = (1.4142 + 5.6568 + 5.831) / 2 = color(red)(6.451)$

Area of triangle base

$B = sqrt(6.451 (6.451 - 1.4142) ( 6.451 - 5.6568) (6.451 - 5.831))$

$color(green)(B ~~ 16)$ sq units

Volume of triangle based pyramid $V = (1/3) B h$ where h is the height of the pyramid.

$color(brown)(V = (1/3) 16 * 6 = 32)$ cubic units

The base of a triangular pyramid is a triangle with corners at (7 ,6 )(7 ,6 ), (4 ,1 )(4 ,1 ), and (3 ,2 )(3 ,2 ). If the pyramid has a height of 6 6 , what is the pyramid's volume?