The base of a triangular pyramid is a triangle with corners at $(2 ,4 )$ , $(3 ,2 )$ , and $(8 ,5 )$ . If the pyramid has a height of $5$ , what is the pyramid's volume?

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Answer 1 · 2016-03-01T06:24:05

$65/6$

The volume of a pyramid is a third of the base area multiplied by the height.

$V = 1/3 xx A xx h$

Refer to this article for more information.

The easiest way to compute the base area is using the Heron's formula.

Using the Pythagorean Theorem, we can get the length of the 3 edges of the triangle.

$l_1 = sqrt{(2 - 3)^2 + (4 - 2)^2} = sqrt5$

$l_2 = sqrt{(2 - 8)^2 + (4 - 5)^2} = sqrt37$

$l_3 = sqrt{(3 - 8)^2 + (2 - 5)^2} = sqrt34$

The semi-perimeter is given by

$s = frac{l_1 + l_2 + l_3}{2} ~~ 7.07$

The area given by Heron's formula is

$A = sqrt{s(s-l_1)(s-l_2)(s-l_3)} = 13/2$

The volume of the pyramid is

$V = 1/3 xx 13/2 xx 5 = 65/6$

The base of a triangular pyramid is a triangle with corners at (2 ,4 )(2 ,4 ), (3 ,2 )(3 ,2 ), and (8 ,5 )(8 ,5 ). If the pyramid has a height of 5 5 , what is the pyramid's volume?