The base of a triangular pyramid is a triangle with corners at $(3 ,8 )$ , $(4 ,9 )$ , and $(5 ,6 )$ . If the pyramid has a height of $7$ , what is the pyramid's volume?

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Answer 1 · 2018-02-16T03:09:31

Volume of pyramid

$V = (1/3)(A_t * h ) ~~ color(purple)(5.1746)$

Given : Triangle corners A (3,8), B(4,9), C(5,6), height of pyramid h = 7.

To find volume of pyramid.

#Using distance formula let’s find the sides of the base triangle.

$vec(AB) = sqrt((4-3)^2+(9-8)^2) = sqrt2 = 1.4142$

$vec(BC) = sqrt((5-4)^2+(6-9)^2) = sqrt10 = 3.1623$

$vec(CA) = sqrt((5-3)^2 + (6-8)^2) = sqrt13 = 3.6056$

Semi perimeter of triangle base

$s = (a + b + c) / 2 = (1.4142 + 3.1623 + 3.6056)/2 ~~ color(blue)(4.09$

Area of base triangle formula, given three sides is

$A_t = sqrt(s(s-a)(s-b)(s-c))$

$=> sqrt(4.09(4.09-3.1623)(4.09-3.6056)(1.4142)) ~~ color (green)(2.2177)$

Volume of pyramid

$V = (1/3)(A_t * h )= (1/3) * 2.2177 * 7 ~~ color(purple)(5.1746)$

The base of a triangular pyramid is a triangle with corners at (3 ,8 )(3 ,8 ), (4 ,9 )(4 ,9 ), and (5 ,6 )(5 ,6 ). If the pyramid has a height of 7 7 , what is the pyramid's volume?