### Why do this problem?

This
problem presents a series of three-dimensional challenges which
encourage the learner to visualise a solid and then use
two-dimensional representations to help them to reach a solution.
On the way to a solution, there are opportunities to practise using
trigonometry and Pythagoras, as well as formulas for volume.

### Possible approach

Each part of this problem could be tackled by small groups who
could then present their solution to other groups. For each
problem, it is best to take some time to visualise what is being
asked, and then draw some diagrams to see what calculations will be
necessary.

The first problem is the simplest of the three, requiring only
visualisation of a right-angled triangle from a section of the
cube, and an application of Pythagoras's theorem.

For the second problem, learners will first need to imagine
how an equilateral triangle could be constructed by cutting through
the cube, and from this work out the dimensions of the largest such
equilateral triangle. The information given is the area of the
triangle so learners will need to come up with a relationship
between the area and the side length.

The third problem is about volume and surface area. In order
to work these out, learners will have to calculate the dimensions
of a tetrahedron cut from the corner of a cube.

### Key questions

What two-dimensional diagrams can be drawn to help to solve
the problems?

If I know the side of an equilateral triangle how can I find
its area?

### Possible extension

### Possible support

Learners who have not met or are not confident with trigonometry
and Pythagoras could solve parts of the problem using scale
drawing.