The base of a triangular pyramid is a triangle with corners at $(3, 1)$ $(3 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(3, 6)$ $(3 ,6 )$ . If the pyramid has a height of $4$ $4$ , what is the pyramid's volume?

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The base of a triangular pyramid is a triangle with corners at $(3, 1)$ $(3 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(3, 6)$ $(3 ,6 )$ . If the pyramid has a height of $4$ $4$ , what is the pyramid's volume?

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Answer 1 · 2018-07-18T13:53:30

Volume of Pyramid $\approx 3.3333$ $~~ 3.3333$ cubic units.

Explanation:

Using distance formula and Heron’s formula we can calculate the base Area of the pyramid. Multiplying it by one third the height will give the volume.

Distance formula $d = \sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(y_{2} - y_{1})}_{2}^{2}}$ $d = sqrt((x_2-x_1)^2 + (y_2 - y_1)^2)$

Heron’s formula $A_{t} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}$ $A_t = sqrt(s * (s-a) * (s-b) * (s-c))$

Where s is the semiperemter of the triangle and a, b, c are the three sides of the base triangle.

Let $A (3, 1), B (2, 7) C (3, 6)$ $A (3,1), B (2,7) C (3,6)$

$a = \sqrt{{(2 - 3)}^{2} + {(7 - 6)}^{2}} = 1.4142$ $a = sqrt((2-3)^2 + (7-6)^2) = 1.4142$

$b = \sqrt{{(3 - 3)}^{2} + {(6 - 1)}^{2}} = 5$ $b = sqrt((3-3)^2 + (6-1)^2) = 5$

$c = \sqrt{{(3 - 2)}^{2} + {(1 - 7)}^{2}} = 6.0828$ $c = sqrt((3-2)^2 + (1-7)^2) = 6.0828$

Semiperemeter $s = \frac{a + b + c}{2} = \frac{12.497}{2} = 6.2485$ $s = (a+b+c)/2 = 12.497/2 = 6.2485$

$A_{t} = \sqrt{6.2485 \cdot (6.2485 - 1.4142) \cdot (6.2485 - 5) \cdot (6.2485 - 6.0828)}$ $A_t = sqrt(6.2485 * (6.2485-1.4142) * (6.2485-5) * (6.2485-6.0828))$

$A_{t} \approx 2.5$ $A_t ~~ 2.5$ sq. units

Volume of pyramid $V_{p} = (\frac{1}{3}) \cdot 2.5 \cdot 4 \approx 3.3333$ $V_p =(1/3) * 2.5 * 4 ~~ 3.3333$ cubic units.

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The base of a triangular pyramid is a triangle with corners at $(3, 1)$ $(3 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(3, 6)$ $(3 ,6 )$ . If the pyramid has a height of $4$ $4$ , 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 (3,1)(3 ,1 ), (2,7)(2 ,7 ), and (3,6)(3 ,6 ). If the pyramid has a height of 44 , 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 $(3, 1)$ $(3 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(3, 6)$ $(3 ,6 )$ . If the pyramid has a height of $4$ $4$ , what is the pyramid's volume?