Volume & Surface Area

This post is about finding volume and surface area of different 3D shapes. A number of the problems are in context and may involve using prior knowledge such as calculating areas or converting units.

Volume of cubes and cuboids

First try to make the biggest cube you can from the 32 unit cubes available using this applet. Drag the red dot on the original cube to position each new cube. Then answer the questions below.

The first two cube numbers are 1 and 8. Can you explain why? What is the third cube number? Can you list the first five cube numbers?

Volume can also be described as capacity, ie the space inside a container. In this problem you might use ideas you have met in science to explain.

In the next two problems, you will need to consider measurements carefully.

And this problem is about how cuboids can be packed inside other cuboids:

• How many packs of butter would fit the box?
• What is the weight of the box when it is full?

Volume of prisms

Here is how this works for a cylinder (the special name for a circular prism), as an example of this:

Here are ten other prisms. Calculate the volume of each one.

Challenge task: The Vases problem requires you to use angles in polygons and trigonometry in order to calculate area and then volume of prisms. There are four vases with the same height:

The bases of the vases are shown above. The perimeter of each base is 24 cm.

• Calculate the area of each base. You may need to use Pythagoras, interior angles of polygons and trigonometry to do this. Give your answers correct to two decimal places.
• Which vase will hold the most water?
• Would you expect a cylindrical vase of circumference 24 cm and the same height as the others to hold more or less water?
• Calculate the volume to check your prediction.

Surface Area

To start thinking about the faces of cubes, the Painted Cube problem starts with a large cube made up of 27 small red cubes as shown in 1. The cube is then dipped into a pot of yellow paint so that the the whole outer surface is covered, as shown in 2.

Then consider if you break the cube up into individual small cubes.

• How many of the small cubes will have yellow paint on their faces?
• Will they all look the same?

Try to describe what the different types of individual cubes there are to someone else. Can you record these results somehow? Then you could try the same thing for a different sized cube and see if you can find any patterns or rules.

In this applet you can change the dimensions of the cuboid (the special name for a rectangular prism) and use the FOCUS and NET sliders to help you work out the surface area, that is the total area of all of the faces.

To explore how we can find the surface area of a cylinder, use the slider to unwrap the cylinder.

Now try to solve these problems

Further ideas

To explore volume and surface areas of other solids such as spheres and cones try out some of the demonstrations in this collection of GeoGebra activities:

Or find out how to pack spheres in this Numberphile clip:

ANSWERS – click below

Volume of cubes and cuboids

The first five cube numbers are 1, 8, 27, 64, 125.

Cuboid Challenge possible solutions here

Volume of prisms
Surface Area

Explore the possible solutions to the Painted Cube problem here.