Home > Geometry > Volume of Solids

The base of a triangular pyramid is a triangle with corners at #(3 ,4 )#, #(2 ,9 )#, and #(5 ,7 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

Answer 1

Sadie Adams

#91/6#

Given

#p_1 = {3,4,0}# #p_2 = {2,9,0}# #p_3 = {5,7,0}#

and defining

#vec u = p_1-p_2# and #vec v = p_3-p_2#

The area spanned by two vectors #vec u, vec v# with the same origin, has a vectorial formula.

#A = 1/2 norm(vec u xx vec v) = 13/2#

The volume of the pyramid is provided by

#V = 1/3 h A = 1/3 cdot 7 cdot 13/2 = 91/6#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Evelyn Arboleda

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

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

First, calculate the area of the base triangle using the given coordinates. 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)| ]

Then, multiply the area of the base triangle by the height of the pyramid, and divide the result by 3 to find the volume.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.