The base of a triangular pyramid is a triangle with corners at $(5 ,8 )$ , $(6 ,7 )$ , and $(2 ,3 )$ . If the pyramid has a height of $2$ , what is the pyramid's volume?

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Answer 1 · 2018-05-21T08:46:48

$color(maroon)(V = (1/3) * A_t * h = 2.65$ cubic units

Given : $A (5,8), B (6,7), C (2,3)$ , h = 2#

Using distance formula we can calculate the lengths of sides a, b, c.

$d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)$

$a = sqrt((6-2)^2+(7-3)^2) = sqrt32 = 5.66$

$b = sqrt((5-2)^2+(8-3)^2) = sqrt34 = 5.83$

$c = sqrt((5-6)^2+(8-7)^2) = sqrt2 = 1.41$

Semi perimeter of base triangle $s = (a+b+c)/2$

$s = (5.66 + 5.83 + 1.41)/2 = 12.9/2 = 6.45$

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

$A_t = sqrt(6.45 * (6.45-5.66)*(6.45-5.83)*(6.45-1.41))$

$A_t = 3.97$

Volume of pyramid $V = (1/3) * A_t * h$

$color(maroon)(V = (1/3) * 3.97 * 2 = 2.65$ cubic units

The base of a triangular pyramid is a triangle with corners at (5 ,8 )(5 ,8 ), (6 ,7 )(6 ,7 ), and (2 ,3 )(2 ,3 ). If the pyramid has a height of 2 2 , what is the pyramid's volume?