Discover the Volume of a Tetrahedron Through Geometric Analysis
Explore the calculation of a tetrahedron's volume using points A, B, C, and D in a 3D space. Understand the geometric relationships, vector operations, and invariant properties that lead to the determination of the tetrahedron's volume. Follow the step-by-step process provided to grasp the concept effectively.
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Presentation Transcript
Volume of Tetrahedron Points A, B, C and D form the vertices of a tetrahedron. A ( -9 , B ( -5 , C ( -6 , D ( -2 , D 7 5 3 5 , , , , 1 1 ) ) ) ) C 17 13 A B a What is the volume of the tetrahedron? SIC_29 Hints: Determine the area of triangle ABC Determine the distance of point D from plane ABC Use volume of pyramid = 1 3 x area of base x perpendicular height
D Determine the vectors ?? and ?? Determine the angle between them using: C ??.?? ?? ?? (dot product) cos? = A B Then area = 1 2??sin? Next, find the unit vector perpendicular to plane ABC, ?. (This could use some inspired detective work if you do not know the cross product.) The perpendicular distance is given by ??. ? Now simply determine the volume.
D Why are all your answers the same, you all had different sets of coordinates? C Compare coordinates and vectors to see if you can spot any similarities A B 4 4 2 and ?? = in all cases! ?? = 2 0 4 Closer inspection will show that all the points A and B lie on the same straight line. Similarly, all the points C and D lie on their own straight line.
D This highlights the fact that if two line segments on different straight lines in 3D are connected to form a tetrahedron, the volume of the tetrahedron is invariant as the line segments slide along the lines. C A B Same volume It s analogous to these triangles having the same area
RESOURCES The resources are in the spreadsheet
