A triangular pyramid has a height of $x$ $x$ , a side length of $2 \sqrt{3}$ $2sqrt3$ , and a volume of $7 \sqrt{3}$ $7sqrt3$ . What are the steps in order to find the height of the pyramid?

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A triangular pyramid has a height of $x$ $x$ , a side length of $2 \sqrt{3}$ $2sqrt3$ , and a volume of $7 \sqrt{3}$ $7sqrt3$ . What are the steps in order to find the height of the pyramid?

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Answer 1 · 2015-12-31T00:43:46

The height of the pyramid is 7

Explanation:

1) The formula of the pyramid's volume is $V = \frac{S_{b a s e} \cdot h}{3}$ $V = (S_(base) * h)/3$

2) Since the base is a equilateral triangle, $tan (60^{o}) = \frac{h e i g h t_{△}}{\frac{s i d e}{2}}$ $tan(60^o)=(height_(triangle))/((side)/2)$
Then $h e i g h t_{△} = (\frac{\sqrt{3}}{2}) \cdot s i d e$ $height_(triangle) = (sqrt(3)/2)*side$

3) The $A r e a_{△} = S_{b a s e} = \frac{b a s e \cdot h e i g h t_{△}}{2}$ $Area_(triangle) = S_(base) = (base*height_(triangle))/2$
(Base of a equilateral triangle is equal to side, since all the sides are equal)

From (2) and (3) we get
$S_{b a s e} = \frac{s i d e \cdot (\frac{\sqrt{3}}{2}) \cdot s i d e}{2} = (\frac{\sqrt{3}}{4}) \cdot s i d e^{2}$ $S_(base) = (side*(sqrt(3)/2)*side)/2 = (sqrt(3)/4)*side^2$

From (1) and the previous formula of the triangle's area, we get
$V = \frac{(\frac{\sqrt{3}}{4}) \cdot s i d e^{2} \cdot h_{p y r a m i d}}{3} = (\frac{\sqrt{3}}{12}) \cdot s i d e^{2} \cdot h_{p y r a m i d}$ $V = ((sqrt(3)/4)*side^2*h_(pyramid))/3 = (sqrt(3)/12)*side^2*h_(pyramid)$
So $h_{p y r a m i d} = (\frac{12}{\sqrt{3}}) \cdot \frac{V}{s i d e^{2}}$ $h_(pyramid)=(12/sqrt(3))*V/(side^2)$

Now we only need to input the known values
$h_(pyramid)=(12/cancel(sqrt(3)))*(7*cancel(sqrt(3)))/(2*sqrt(3))^2 = (12*7)/(4*3) = 7$

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A triangular pyramid has a height of $x$ $x$ , a side length of $2 \sqrt{3}$ $2sqrt3$ , and a volume of $7 \sqrt{3}$ $7sqrt3$ . What are the steps in order to find the height of the pyramid?

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Explanation:

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A triangular pyramid has a height of $x$ $x$ , a side length of $2 \sqrt{3}$ $2sqrt3$ , and a volume of $7 \sqrt{3}$ $7sqrt3$ . What are the steps in order to find the height of the pyramid?