The base of a triangular pyramid is a triangle with corners at $(2 ,6 )$ , $(5 ,4 )$ , and $(7 ,5 )$ . If the pyramid has a height of $7$ , what is the pyramid's volume?

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Answer 1 · 2018-06-09T02:04:00

$color(green)("Volume of Pyramid " V_p = (1/3) * A_b * h = 8.29 " cubic units"$

$color(violet)("Volume of Pyramid " V_p = (1/3) * A_b * h$

$Area of base triangle " A_b = sqrt(s (s-a) (s-b) (s-c)), " using Heron's formula$

$A(2,6), B(5,4), C(7,5), h = 7$

$a = sqrt((5-7)^2 + (4-5)^2) = 2.24$

$b = sqrt((7-2)^2 + (5-6)^2) = 5.1$

$c = sqrt((2-5)^2 + (6-4)^2) = 3.61$

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

$s = (2.24 + 5.1 + 3.61) / 2 ~~ 5.48$

$A_b = sqrt(5.48 * (5.48 - 2.24) * (5.48 - 5.1) * (5.48 - 3.61)) = 3.55$

$color(green)("Volume of Pyramid " V_p = (1/3) * A_b * h = (1/3) * 3.55* 7 = 8.29 " cubic units"$

The base of a triangular pyramid is a triangle with corners at (2 ,6 )(2 ,6 ), (5 ,4 )(5 ,4 ), and (7 ,5 )(7 ,5 ). If the pyramid has a height of 7 7 , what is the pyramid's volume?