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?

Question

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?

Ella Armstrong · Accepted Answer

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)

The base of the altitude's coordinates are (7,1), which we obtain by solving Equations (1) and (2).

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#

Parker Allen · Answer

undefined To find the volume of the triangular pyramid, we can use the formula:

$$ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$

First, we need to find the area of the base triangle. We can use the formula for the area of a triangle given its vertices. Let's label the vertices as $A(2, 6)$, $B(5, 3)$, and $C(8, 2)$.

The base area can be calculated using the formula:

$$ \text{Area} = \frac{1}{2} \times |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $$

Using the coordinates of the vertices, we get:

$$ \text{Area} = \frac{1}{2} \times |2(3 - 2) + 5(2 - 6) + 8(6 - 3)| $$

$$ \text{Area} = \frac{1}{2} \times |2 + (-8) + 18| $$

$$ \text{Area} = \frac{1}{2} \times |12| $$

$$ \text{Area} = 6 \text{ square units} $$

Now, we have the base area and the height of the pyramid. Substituting these values into the formula for volume:

$$ V = \frac{1}{3} \times 6 \times 18 $$

$$ V = 36 \text{ cubic units} $$

Therefore, the volume of the triangular pyramid is $36$ cubic units.