The nets of geometric figures are formed when we extend a 3D figure. In general, a geometric net can be defined as a two-dimensional figure that can be modified to form a three-dimensional figure.
Here, we will look at the geometric nets of cubes, cones, cylinders, tetrahedra and octahedra. In addition, we will look at some of their important characteristics.
Nets of a cube
A cube is a six-sided regular polyhedron. This means that all the faces of a cube are squares. The following is the most common geometric net of a cube:
Additionally, there are also other ways to form geometric nets of a cube. The following are all possibilities. Note that if we fold any of these nets, we can form a cube.
Nets of a cone
A cone is a three-dimensional figure that has a circular base and a lateral surface with a pointed top. This means that the geometric net must include a circular base and a curved surface. In the following animation, we can see how the geometric net of a cone is formed.
Depending on the relationship between the radius and the height of the cone, we can obtain three variations of this net, which are shown in the following diagram.
To determine which of the three nets corresponds to a given cone, we have to compare the length of the radius with the length of the slant height of the cone. Then, we have the following:
1. If the slant height, represented with l, has a length equal to 2r (equivalent to the diameter), the cone will form the following geometric net:
2. If the slant height has a length less than 2r, the cone will form the following geometric net:
3. If the slant height has a length greater than 2r, the cone will form the following geometric net:
Nets of a cylinder
A cylinder is a three-dimensional figure made up of two circular bases and a lateral surface that connects the two bases. The geometric net of a cylinder contains two circles that form the two circular bases and a rectangle, which forms the curved surface when folded.
In the following animation, we can see how the geometric net of a cylinder is formed.
We can observe that the bases remain unchanged. The radius and therefore the area of the circular bases is the same in the cylinder and in its 2D net.
However, the curved surface of the cylinder is extended to form a rectangle. The height of the rectangle is equal to the height of the cylinder.
The length of the rectangle corresponds to the circumference of the circular bases. Therefore, the length of the rectangle is equal to 2πr, where r is the radius of the bases.
Nets of a tetrahedron
Tetrahedra are three-dimensional figures formed by four triangular faces. One triangular face is the base and the other three faces form the lateral surface, meeting at the top vertex. This means that the net of a tetrahedron must include four triangular faces.
In the following animation, we can see how the geometric net of a tetrahedron is folded to form the three-dimensional tetrahedron.
We can see that the base of the tetrahedron remains unchanged. Each lateral face bends at each base edge until connecting at the top vertex.
When we talk about tetrahedrons, we usually mean a regular tetrahedron, so all four faces will be congruent equilateral triangles.
Nets of an octahedron
Octahedra are one of the five Platonic solids. These three-dimensional figures have eight congruent triangular faces. That is, the faces have the same shape and the same dimensions.
This means that the geometric net of an octahedron must contain eight congruent triangular faces.
In the following animation, we can see how the net of an octahedron can be folded to form a three-dimensional octahedron.
Since the octahedron consists of two square pyramids joined at their base, we can fold each of these pyramids separately. In the diagram, we see that the lateral faces of each pyramid are folded around the faces that act as temporary bases.
When we talk about octahedrons, we usually mean a regular octahedron, so all eight faces will be congruent equilateral triangles.
Interested in learning more about geometric figures? Take a look at these pages: