Finding the Volume of a tetrahedron using Spherical Coordinates

In summary, to find the volume of a tetrahedron under a plane with equation 3x + 2y + z = 6 and in the first octant, use spherical coordinates with the following equations: x = psin(phi)cos(theta), y = psin(phi)sin(theta), and z = pcos(phi). The boundaries for p are 0 and (6/(cos(phi)+sin(phi)(3cos(theta)-2sin(theta)). The boundaries for theta are 0 and pi/2. And the boundaries for phi are 0 and pi/2. The resulting volume is six.
  • #1
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Find the volume of a tetrahedron under a plane with equation 3x + 2y + z = 6 and in the first octant. Use spherical coordinates only. The answer is six.




x=psin(phi)cos(theta)
y=psin(phi)sin(theta)
z=pcos(phi)



I've been trying to figure out the boundaries of this particular problem all night. To be honest, I'm completely at a loss. I have p going between 0 and (6/(cos(phi)+sin(phi)(3cos(theta)-2sin(theta)). I believe that theta is between 0 and pi/2, although I'm not entierly sure on that one. As far as phi goes I believe the upper limit is pi but the lower limit is a mystery to me.
 
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  • #2
[itex]\phi[/itex] would go from 0 to [itex]\pi/2[/itex] as does [itex]\theta[/itex].
 

Related to Finding the Volume of a tetrahedron using Spherical Coordinates

What is a tetrahedron?

A tetrahedron is a three-dimensional shape with four triangular faces, six edges, and four vertices. It is a type of pyramid and is one of the five platonic solids.

What are spherical coordinates?

Spherical coordinates are a method for locating points in three-dimensional space using a radius, an angle, and a height. The radius is the distance from the origin, the angle is measured from a fixed reference direction, and the height is the distance from the xy-plane.

How do you find the volume of a tetrahedron using spherical coordinates?

To find the volume of a tetrahedron using spherical coordinates, you can use the formula V = (1/3) * a^3, where a is the length of one of the edges of the tetrahedron. You can also use the formula V = (1/6) * a^2 * h, where a is the length of one of the edges and h is the height of the tetrahedron.

Why use spherical coordinates to find the volume of a tetrahedron?

Spherical coordinates are useful for finding the volume of a tetrahedron because they allow you to take into account the angle and height of the shape, rather than just the length of the edges. This can be particularly helpful for more complex shapes where the edges may not all have the same length.

Are there any limitations to using spherical coordinates for finding the volume of a tetrahedron?

One limitation of using spherical coordinates for finding the volume of a tetrahedron is that it assumes the shape is a perfect tetrahedron with four equal edges. If the shape is irregular or has unequal edges, this method may not provide an accurate result. Additionally, this method may be more complex and time-consuming than other methods for finding the volume of a tetrahedron.

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