The base of a triangular pyramid is a triangle with corners at $(2 ,6 )$ , $(5 ,3 )$ , and $(8 ,2 )$ . If the pyramid has a height of $18$ , what is the pyramid's volume?

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Answer 1 · 2017-11-26T12:52:23

Volume of the pyramid with triangular base is $18$

Volume of triangular pyramid = $(1/3) AH$ where A is the area of the triangular base and H is the height of the pyramid.

Area of triangular base = $(1/2) b h$ where b is the base and h is the height of the triangle.

Base $= sqrt((5-2)^2 + (3-6)^2) = sqrt(3^2 + 3^2 )=color(blue)( 3 sqrt2)$

Eqn of Base =is
$((y-y_1)/(y_2-y_1)) = ((x-x_1)/(x_2-x_1))$

$((y-6)/(3-6))=((x-2)/(5-2))$

$(y-6) =( -x + 2)$
$y+x = 8$ Eqn. (1)
Slope of base $m = (y_2 - y_1) / (x_2 - x_1)$
Slope $m = (3-6)/(5-2) = -1$

Slope of altitude $m_1 = -(1/m) = -(1/(-1)) = 1$

Eqn of Altitude is
$(y-y_3) = m_1(x-x_3)$

$y- 2 = 1 (x-8)$
$y-x = -6$ . Eqn (2)

Solving Eqns (1) & (2), we get coordinates of the base of the altitude.
Coordinates of base of altitude are (7,1)

Height of altitude $= sqrt((8-7)^2 + (2-1)^2 )= color(red)(sqrt2)$

Area of triangular base $= (1/2) color(blue)(3 sqrt2) color(red )(sqrt2)$

Area $=3$

Volume of pyramid = $(1/3) * 3 * 18 = 18$

The base of a triangular pyramid is a triangle with corners at (2 ,6 )(2 ,6 ), (5 ,3 )(5 ,3 ), and (8 ,2 )(8 ,2 ). If the pyramid has a height of 18 18 , what is the pyramid's volume?