How do you find the volume of a tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 4 cm 4 cm and 2 cm?

Answer 1

If this is a homework assignment, please consider trying it yourself first, otherwise the value of the reinforcement to the lesson offered by the homework will be lost on you.

To find the volume of a tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 4 cm 4 cm and 2 cm, start by visualizing the tetrahedron.

If three edges are mutually perpendicular, they could be the X, Y, and Z axes. Similarly, if three faces are mutually perpendicular, they could be the X-Y, X-Z, and Y-Z axis planes.

The four vertices are at (0,0,0), (4,0,0), (0,4,0), and (0,0,2) in three space, coordinates given as (x,y,z). The fourth face is contained in the plane containing the last three coordinates.

This is a simple problem in integration, adding up the volume of triangles of thickness dx, as dx approaches zero. (You could pick any axis to integrate - dx, dy, or dz - it does not matter - lets pick dx.)

The height of the triangular section in the Y-Z plane is y. As a function of x, y = 4-x. STOP. If you do not understand that, draw the X-Y plane at z=0. It is a triangle, and the hypotenuse goes from (0,4) to (4,0). That is slope (m) = -1 and y intercept (b) = 4.

The base of the triangular section in the Y-Z plane is z. As a function of x, z = 2-x/2. Again, STOP and understand that. Draw the X-Z plane at y=0. Its the same deal, just swap y for z.

OK. We know the area of a triangle is one half base times height. In this case, slicing parallel to the Y-Z plane, the area is 1/2 (4-x) (2-x/2). Simplifying, that is a(x) = 1/8 x2 - x + 2. STOP. That is simple algebra, but work it out to reinforce the calculation.

Now, to compute the volume of the tetrahedron, determine the definite integral of a(x) dx over the range 0 to 4.

That is the definite integral of (1/8 x2 - x + 2) dx from 0 to 4, which is 1/32 x3 - 1/2 x2 + 2 x from 0 to 4, which is 2. Again, STOP. Make sure you know how to integrate a polynomial of degree two. and convert the integral from indefinite to definite.

Since the original units were in cm, the volume of the tetrahedron is 2 cm3.