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

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Answer 1 · 2018-03-26T08:12:49

$5.67$ $5.67$ cubic units

First Of All, Find the Area of the Triangular Base.

So, The Area of the Base of The Pyramid :-

$\frac{1}{2} {x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})}$ $1/2{x_1(y_2 -y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)}$

[If The Coordiantes for the vertices of the triangles are $(x_{1}, y_{1}),$ $(x_1,y_1),$

$(x_{2}, y_{2}) and (x_{3}, y_{3})$ $(x_2,y_2) and (x_3,y_3)$ respectively.]

$= \frac{1}{2} {2 (3 - 2) + 5 (2 - 7) + 8 (7 - 3)}$ $= 1/2{2(3-2) + 5(2-7) + 8(7 - 3)}$ sq. units

$= \frac{1}{2} {2 - 25 + 40}$ $= 1/2{2 -25 +40}$ sq. units

$= \frac{1}{2} (17)$ $= 1/2(17)$ sq. units

$= 8.5$ $= 8.5$ sq. units.

So, The Volume of The Triangular Pyramid

$= \frac{1}{3} (a h)$ $= 1/3(ah)$ [Where $a$ $a$ is the area of the base and $h$ $h$ is the height of the pyramid.]

$= \frac{1}{3} (8.5 \cdot 2)$ $= 1/3 (8.5*2)$ cubic units

$= \frac{17}{3}$ $= 17/3$ cubic units

$= 5.67$ $=5.67$ cubic units (approx.)

Hope this helps.

The base of a triangular pyramid is a triangle with corners at (2 ,7 )(2,7)(2 ,7 ), (5 ,3 )(5,3)(5 ,3 ), and (8 ,2 )(8,2)(8 ,2 ). If the pyramid has a height of 2 22 , what is the pyramid's volume?