The base of a triangular pyramid is a triangle with corners at #(7 ,8 )#, #(2 ,3 )#, and #(8 ,3 )#. If the pyramid has a height of #6 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(7  ,8  )#, #(2  ,3   )#, and #(8  ,3  )#.  If the pyramid has a height of #6  #, what is the pyramid's volume?

Violet Aldrich · Accepted Answer

Volume of pyramid is #30.0# cubic unit

Triangle's sides are #A=sqrt((7-2)^2+(8-3)^2) =sqrt50=7.07# #B=sqrt((2-8)^2+(3-3)^2) =sqrt36=6.0# #A=sqrt((8-7)^2+(3-8)^2) =sqrt26=5.1:.#Semi perimeter:#s=(A+B+C)/2=9.09:.# Base area: #A_b = sqrt(s(s-a)(s-b)s-c))=sqrt(9.09(9.09-7.07)(9.09-6)(9.09-5.1))=15.0:.#Volume of pyramid: #v=1/3*A_b*ht =1/3*6*15=30.0# cubic unit[Ans]

Sadie Adams · Answer

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

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

First, calculate the base area of the triangular pyramid. You can use the formula for the area of a triangle given its vertices:

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

where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the vertices of the triangle.

Substitute the given coordinates into the formula:

$$ A = \frac{1}{2} \times |7(3 - 3) + 2(3 - 8) + 8(8 - 3)| $$

$$ A = \frac{1}{2} \times |0 + 2(-5) + 8(5)| $$

$$ A = \frac{1}{2} \times |0 - 10 + 40| $$

$$ A = \frac{1}{2} \times |30| $$

$$ A = 15 $$

Now, you have the base area ($A$) and the height of the pyramid ($h = 6$). Plug these values into the volume formula:

$$ V = \frac{1}{3} \times 15 \times 6 $$

$$ V = \frac{1}{3} \times 90 $$

$$ V = 30 $$

So, the volume of the triangular pyramid is $30$ cubic units.