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

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Answer 1 · 2018-05-17T06:46:57

Volume of triangular base pyramid $color(violet)(V = 466.47$ cub. units

$A (2,4), B (3,2), C (5,5)$

Using distance formula,

$bar (AB) = c = sqrt((3-2)^2 + (2-4)^2) = sqrt5 = 2.235$

$bar (BC) = a = sqrt((5-2)^2 + (5-4)^2) = sqrt 10 = 3.162$

$bar (AC) = b = sqrt((3-5)^2 + (2-5)^2) = sqrt 13= 3.606$

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

Semi perimeter $s (a -+ b+ c) / 2 = (2.235 + 3.162 + 3.606)/2 = 18$

$A_b = sqrt(18* 15.765 * 16.838 * 16.394) = 279.88$

Formula for volume of pyramid $V = (1/3) * A_b * h$

$color(violet)(V = (1/3) * 279.88 * 5 = 466.47$ cub. units.

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