### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

### Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

# Three Cubes

### 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

This printable worksheet may be useful: Three Cubes.

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 support

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

### Possible extension

The Spider and the Fly gives another opportunity to visualise a problem in three dimensions.