### Qqq..cubed

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

### When the Angles of a Triangle Don't Add up to 180 Degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.

### The Spider and the Fly

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

# Tin Tight

##### Stage: 4 Challenge Level:

How many cubic centimetres make a litre?

If the tin (cylinder) had a diameter of $10 \; \text{cm}$ what would the height be?

What surface area would this tin have? Don't forget the top and the bottom.

Would a diameter of $20 \; \text{cm}$ have a greater or lesser surface area?

Can you write out a procedure to follow for any diameter you have to try?

Try to follow your procedure and use trial and improvement to get increasingly close to the diameter which gives the lowest value for the complete surface area.

What was the height for that radius value?

What was the proportion between the height and the diameter?

A spreadsheet might be useful once you are sure of your method.