The base of a triangular pyramid is a triangle with corners at #(2 ,5 )#, #(6 ,9 )#, and #(3 ,8 )#. If the pyramid has a height of #15 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(2 ,5  )#, #(6  ,9   )#, and #(3  ,8  )#.  If the pyramid has a height of #15  #, what is the pyramid's volume?

Sadie Allen · Accepted Answer

#20.001#

To find volume of a triangular pyramid of height #15# and base a triangle with corners at #A(2,5)#, #B(6,9)#, and #C(3,8)# we must find the area of the base triangle.

The triangle's sides are located as follows.

#AB=sqrt((6-2)^2+(9-5)^2)=sqrt32=5.6568#

#BC=sqrt((6-3)^2+(9-8)^2)=sqrt(9+1)=sqrt10=3.1623#

#CA=sqrt((3-2)^2+(8-5)^2)=sqrt(1+9)=sqrt10=3.1623#

Using Heron's formula #s=(5.6568+3.1623+3.1623)/2=5.9907#

and area of triangle is #sqrt(5.9907xx(5.9907-5.6568)xx(5.9907-3.1623)xx(5.9907-3.1623)# i.e. #sqrt(5.9907xx0.3339xx2.8284xx2.8284)=sqrt16.0021=4.0002# (approx.)

As volume of pyramid #1/3xxheightxxarea of base#, it is #1/3xx4.0002xx15=20.001#

Autumn Armstrong · Answer

undefined To find the volume of a 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, which is:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

We'll first find the lengths of the sides of the base triangle using the distance formula:

$$ \text{Side 1} = \sqrt{(6-2)^2 + (9-5)^2} $$
$$ \text{Side 2} = \sqrt{(3-6)^2 + (8-9)^2} $$
$$ \text{Side 3} = \sqrt{(3-2)^2 + (8-5)^2} $$

Then, we'll find the semi-perimeter of the triangle, which is half the sum of its sides:

$$ \text{Semi-perimeter} = \frac{\text{Side 1} + \text{Side 2} + \text{Side 3}}{2} $$

Using Heron's formula, the area of the triangle is:

$$ \text{Area} = \sqrt{\text{Semi-perimeter} \times (\text{Semi-perimeter} - \text{Side 1}) \times (\text{Semi-perimeter} - \text{Side 2}) \times (\text{Semi-perimeter} - \text{Side 3})} $$

Once we have the area of the base triangle and the height of the pyramid, we can plug these values into the formula for the volume of a pyramid:

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

Substituting the calculated values, we can find the volume of the pyramid.