Prove that the volume of a tetrahedron with mutually perpendicular adjacent sides of lengths a, b, and c, is abc/6.
Keep in mind: - Finding the volume of a solid bounded by the 3 coordinate planes and a given plane - 3 noncollinear points make a plane
I was looking at this two ways...
one, I know that I can place the three sides that are mutually perpendicular on the xyz plane so that the point where they intersect is the origin. I labeled the side along the z-axis c, with a,b along the x and y planes respectively. I can see that the the three sides end on noncollinear points, making a plane. But how do I get the equation of that plane??
The second way was through a similar question posted (but instead of a,b and c there were real numbers)... same visulaization but looking for the area of a cross section perpendicular to the xy plane, with height z. The cross section would be a similar triangle and this is where I get lost... why would this cross section be with sides scaled at (c-z)/c? and why is its area (1/2)(a)(b)(c-z/c)^2? I understand that I have to take the integral from z=0 to z=c of (1/2)(a)(b)(c-z/c)^2, but I don't understand how to get to that point. Any help is appreciated. Thank you!