The base of a triangular pyramid is a triangle with corners at $(2, 2)$ $(2 ,2 )$ , $(3, 1)$ $(3 ,1 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $6$ $6$ , what is the pyramid's volume?

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The base of a triangular pyramid is a triangle with corners at $(2, 2)$ $(2 ,2 )$ , $(3, 1)$ $(3 ,1 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $6$ $6$ , what is the pyramid's volume?

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Answer 1 · 2016-03-02T09:29:14

$8$ $8$

Explanation:

To find volume of a triangular pyramid of height $6$ $6$ and base a triangle with corners at $A (2, 2)$ $A(2,2)$ , $B (3, 1)$ $B(3,1)$ , and $C (7, 5)$ $C(7,5)$ we must find the area of the base triangle.

The sides of triangle can be found as follows.

$A B = \sqrt{{(3 - 2)}^{2} + {(1 - 2)}^{2}} = \sqrt{2} = 1.4142$ $AB=sqrt((3-2)^2+(1-2)^2)=sqrt2=1.4142$

$B C = \sqrt{{(7 - 3)}^{2} + {(5 - 1)}^{2}} = \sqrt{16 + 16} = \sqrt{32} = 5.6568$ $BC=sqrt((7-3)^2+(5-1)^2)=sqrt(16+16)=sqrt32=5.6568$

$C A = \sqrt{{(7 - 2)}^{2} + {(5 - 2)}^{2}} = \sqrt{25 + 9} = \sqrt{34} = 5.831$ $CA=sqrt((7-2)^2+(5-2)^2)=sqrt(25+9)=sqrt34=5.831$

Using Heron's formula $s = \frac{1.4142 + 5.6568 + 5.831}{2} = 6.451$ $s=(1.4142+5.6568+5.831)/2=6.451$

and area of triangle is $\sqrt{6.451 \times (6.451 - 1.4142) \times (6.451 - 5.6568) \times (6.451 - 5.831)}$ $sqrt(6.451xx(6.451-1.4142)xx(6.451-5.6568)xx(6.451-5.831)$
i.e. $\sqrt{6.451 \times 5.0368 \times 0.7942 \times 0.62} = 4$ $sqrt(6.451xx5.0368xx0.7942xx0.62)=4$ (approx.)

As volume of pyramid $\frac{1}{3} \times h e i g h t \times a r e a o f b a s e$ $1/3xxheightxxarea of base$ , it is $\frac{1}{3} \times 4 \times 6 = 8$ $1/3xx4xx6=8$

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The base of a triangular pyramid is a triangle with corners at $(2, 2)$ $(2 ,2 )$ , $(3, 1)$ $(3 ,1 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $6$ $6$ , what is the pyramid's volume?

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Explanation:

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The base of a triangular pyramid is a triangle with corners at (2 ,2 )(2,2)(2 ,2 ), (3 ,1 )(3,1)(3 ,1 ), and (7 ,5 )(7,5)(7 ,5 ). If the pyramid has a height of 6 66 , what is the pyramid's volume?

1 Answer

Explanation:

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The base of a triangular pyramid is a triangle with corners at $(2, 2)$ $(2 ,2 )$ , $(3, 1)$ $(3 ,1 )$ , and $(7, 5)$ $(7 ,5 )$ . If the pyramid has a height of $6$ $6$ , what is the pyramid's volume?