Four points on a cube
What is the surface area of the tetrahedron with one vertex at O
the vertex of a unit cube and the other vertices at the centres of
the faces of the cube not containing O?
Age
16 to 18
Challenge level
Problem
Take a cube of side length one unit. A vertex of the cube and the centres of the three faces not containing that vertex form the vertices of a tetrahedron. What is the surface area of this tetrahedron?
Image
Getting Started
3D coordinates are useful.
Student Solutions
Congratulations to Ang Zhi Ping from River Valley High School, Singapore for this solution.
Take the vertex of the cube at the origin $(0,0,0)$ as shown in the diagram. The tetrahedron has vertices $O,X,Y,Z$, where the three centres $X$, $Y$ and $Z$ are as follows:
Image
The area required is the sum of the areas of the triangles $XYZ$ (which is equilateral), and $OXY$, $OYZ$, $OZX$ (which are congruent to each other and isosceles).
Now $XY= 1/\sqrt{2}$ so that
Teachers' Resources
Why do this
problem?
It gives experience of using 3D coordinates, visualisation, and using the distance formula in 3D, and formulae for the area of a triangle.
Possible approach
A good diagram isessential. You might discuss and draw and label the diagram as a class effort and then the students could complete the calculatons independently.
Key questions
What are the coordinates of the vertices of the tetrahedron?
It gives experience of using 3D coordinates, visualisation, and using the distance formula in 3D, and formulae for the area of a triangle.
Possible approach
A good diagram isessential. You might discuss and draw and label the diagram as a class effort and then the students could complete the calculatons independently.
Key questions
What are the coordinates of the vertices of the tetrahedron?