Nets of Cubes

What are Nets?

A net is a two-dimensional representation of a three-dimensional solid. It shows all the faces of the solid laid out flat, connected along their edges. When folded along these edges, the net forms the solid.

Nets of a Cube

A cube has six identical square faces. The net of a cube must consist of six squares arranged in a way that they can be folded to create the cube.

There are 11 distinct ways to arrange six squares to form a cube's net. Some examples include:

A cross-shaped arrangement with four squares in a line and two squares attached on either side.
A T-shaped arrangement with one square at the top and three squares in a line below it, and two squares attached to either side of the middle square.

Applications of Cube Nets

Understanding Surface Area: By analyzing the net, the surface area of a cube can be calculated as the sum of the areas of its six square faces.

\text{Surface Area} = 6 \times (\text{side length})^2

Practical Construction: Nets are used in packaging design and model making.
Visualization Skills: Working with nets improves spatial reasoning.

Worked Examples

infoNote

Example 1: Calculate the Surface Area of a Cube Using a Net

Problem: A cube has a side length of $4 \, \text{cm}$

Find its surface area.

Solution:

Step 1: Each square face has an area of:

\text{Area} = (\text{side length})^2 = 4^2 = :highlight[16 \, \text{cm}^2]

Step 2: Multiply by $6$ (the number of faces):

\text{Surface Area} = 6 \times 16 = :success[96 \, \text{cm}^2]

Answer: The surface area is $96 \, \text{cm}^2$

infoNote

Example 2: Identify a Valid Net of a Cube

Problem: Which of the following arrangements of six squares can fold into a cube?

A straight line of six squares.
A cross-shaped net with four squares in a line and one square attached to each of the first and last squares.
A T-shaped net as described above.

Solution:

A straight line cannot form a cube as the squares cannot fold to enclose the space.
The cross-shaped and T-shaped arrangements are valid nets for a cube.

Answer: Options $2$ and $3$ are valid cube nets.

Summary

Nets Definition: A flat, two-dimensional pattern that can be folded into a three-dimensional solid.
Cube Nets: Must consist of six squares; there are $11$ distinct valid arrangements.
Surface Area Calculation: The sum of areas of all six square faces: $Surface Area=6×(side length)2\text{Surface Area} = 6 \times (\text{side length})^2$
Applications include packaging, model design, and enhancing spatial visualization skills. Explore different cube nets to strengthen spatial reasoning!

Nets of Cubes Simplified Revision Notes for Leaving Cert Mathematics