The base of a triangular pyramid is a triangle with corners at #(8 ,5 )#, #(6 ,2 )#, and #(5 ,9 )#. If the pyramid has a height of #8 #, what is the pyramid's volume?
The volume is
The area of the base times the height equals the volume:
It is best to use a determinant to compute the area since we are given three points.
The determinant that will be used to calculate the area given three points is as follows:
Changing the three points to:
#A = +-1/2 |(8,5,1), (6,2,1), (5,9,1)|#
#A = +-1/2 | (8,5,1,8,5), (6,2,1,6,2), (5,9,1,5,9) |#
Add the results of multiplying each of the major diagonals:
#A = +-1/2 | (color(red)(8),color(green)(5),color(blue)(1),8,5), (6,color(red)(2),color(green)(1),color(blue)(6),2), (5,9,color(red)(1),color(green)(5),color(blue)(9)) | = #
The minor diagonals should be multiplied and then subtracted from the total of the major diagonals.
#A = +-1/2 | (8,5,color(red)(1),color(green)(8),color(blue)(5)), (6,color(red)(2),color(green)(1),color(blue)(6),2), (color(red)(5),color(green)(9),color(blue)(1),5,9) | =#
By signing up, you agree to our Terms of Service and Privacy Policy
To find the volume of the triangular pyramid, you can use the formula:
( V = \frac{1}{3} \times A_{\text{base}} \times h )
Where ( A_{\text{base}} ) is the area of the base of the pyramid and ( h ) is the height of the pyramid.
First, you need to calculate the area of the base of the pyramid using the coordinates of the triangle's vertices. Then, you can substitute the values into the formula to find the volume.
Using the coordinates provided, you can calculate the area of the base using the formula for the area of a triangle given its vertices.
Then, substitute the calculated area of the base and the given height into the formula for the volume of the pyramid to find the final result.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
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.
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.
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.
- A closed rectangular box has a height of 2 feet, a length of 4 feet, and a width of 4 feet. What is the volume of the box? What is the surface area of the box?
- An ellipsoid has radii with lengths of #9 #, #11 #, and #4 #. A portion the size of a hemisphere with a radius of #8 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?
- How do you find diameter of a circle with circumference?
- A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #4 # and #1 # and the pyramid's height is #7 #. If one of the base's corners has an angle of #(5pi)/6#, what is the pyramid's surface area?
- Cups A and B are cone shaped and have heights of #35 cm# and #23 cm# and openings with radii of #14 cm# and #7 cm#, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7