### 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?

# Changing Areas, Changing Volumes

### Why do this problem?

Working on this problem will give students a deeper understanding of the relationship between volume and surface area, and how they change as the dimensions of a cuboid are altered.

### Possible approach

This problem follows on from Changing Areas, Changing Perimeters. We suggest students start with the rectangles task from that problem to introduce them to the structure of the grid they will be using here.

Once students are familiar with the grid structure, hand out these cards and invite students to work in pairs to arrange the cards in a grid like this:

Students could use multilink cubes or draw each cuboid on isometric paper to support them as they work on the task.

For those who finish quickly, ask them the question from the problem about extending the grid like this:

Towards the end of the lesson, bring the class together to share any efficient strategies they used to complete the task.

Pose the question "If I know two cuboids have the same volume, how can I decide, just by looking at their dimensions, which has the greater surface area?"

Draw out students' ideas about the properties of long and thin cuboids as opposed to those that are almost cubes.
(This is the three dimensional analogue of short and fat rectangles having a smaller perimeter than long thin ones, when their areas are equal.)

Finally, discuss the possible content of the four extra spaces in the extended grid and strategies they used to generate possibilities.

### Possible extension

Challenge students to design their own set of nine cards that can be arranged in this way. If students are restricted to whole numbers it is quite challenging to create cuboids with equal surface areas.

### Possible support

Changing Areas, Changing Perimeters provides a two-dimensional version of this three-dimensional problem.