V = 12

Introduction

Each multi-link cube have a volume of 1 unit3.  

1. Get 12 multi-link cubes.  How many boxes (cuboids - click me if you are unsure what I am) with a volume of 12 units3 can you make?  

   • How can you describe the cuboid you have made so that other people can make it?

   • How can you be sure that you have found them all?

   • How can you list your cuboids out in a systematic way? Hint: think table.

2. Draw all of them on isometric paper either on a hard copy (PRINT ME) or on Desmos.  

   • Look at this video to learn how to do this.

3. What is the respective surface area of each cuboid? Click on the hyperlink if you are unsure how to find the surface area of a cuboid.

   • You may want to use this from NCTM (Firefox works better than Chrome) to check your answers for surface area is correct.

   • What do you notice about the shapes with the largest surface area?

   • What do you notice about the shapes with the smallest surface area?

4. Now repeat the process (steps 1-3) with 18 cubes. 

   • Do you think you will be able to make more or fewer boxes as you have more cubes? 

   • How many more/ fewer boxes?

5. Can you repeat the process (steps 1-3) with 20 cubes BUT without actually building the boxes?

6. Can you generalise rules for the volume and surface area?

Further Questions and Challenges

1. 12, 18 and 20 cubes all produce 4 cuboids, what is the smallest number of cubes required to produce more than 4 cuboids?

2. If you have a bigger volume (number of cubes), does it mean you will always be able to get more boxes?  Why/ why not?

3. Now try different volumes (number of cubes). 

   • Which numbers of cubes give the greatest number of different cuboids?

   • Which numbers of cubes give the fewest number of different cuboids?

   • What do you notice about these answers?

4. What are the dimensions and volumes of cuboids if the surface area is

   • 30?

   • 46?

5. Is there a connection between the dimensions of the box and the surface area?  

6. How do we maximise/minimise surface area for a given volume?

7. Can you find cuboids where the volume and surface area are numerically equal?  Also try this Transum task!

Further Practice

Some of the essential skills introduced in this lesson are "drawing on isometric paper", "surface area of a cuboid" and "volume of a cuboid".  You may also wish to find out how to find surface area and volume of other 3D shapes.  The relevant skills can be found on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.  Watch any video and/or go through any online lesson as you see fit.  

Transum

• Surface Area - work out the surface areas of common solid shapes in this collection of exercises.  Only the first 4 levels are appropriate.

• Volume: Use formulae to solve problems involving the volumes of cuboids, cones, pyramids, prisms and composite solids.  Only the first 3 levels are appropriate.

• Volume Equals Surface Area: Find the cuboids with integer side lengths where the volume is numerically equal to the surface area.

For more goodies on 3D shapes on transum, click on the hyperlink. 

nRich

Try these tasks on nRich to do with surface area and volume of cuboids.  For a taste, try this Changing Areas, Changing Volumes task.