The envelope of a triangular pyramid


A discussion question: Which pictures of solids A or B are polyhedra? Explain.

POLYHEDRON: A 3 dimensional solid formed by polygons.


Answer: A polyhedron is a 3 dimensional solid with all of the faces being polygons. Only the solids seen in group A are polyhedrons.


Each pupil chooses three sheets of origami paper.


Folding instructions: Fold opposite vertices on each sheet.


A discussion question: Which type of triangle is made after folding? Explain.


Answer: Two sides of the triangle are equal in length and the triangle is named an isoceles triangle.
The triangle has also one right angle and it is named a right angled isoceles triangle.


Folding instructions: Fold the triangle as shown.

At this stage it is important that the pupils make their folds pressed down well.


Folding instructions: Place the three triangles as shown in the animation. Make sure that the longest sides are on the table.


Folding instructions:Place the three triangles as shown in the animation. Make sure that the longest sides are on the table.


Folding instructions: Join the three triangles in the way shown so that only one color is seen on each face.

Check that the folding shows only three colors.


Folding instructions:The three triangles have been joined.


A discussion question: What shape is seen? Explain.


Answer: The joined triangles have formed the envelope of a triangular pyramid constructed from triangles having a common vertex.


A discussion question: Why is it called the envelope of a pyramid and not a pyramid?


Answer: It is named the envelope of a triangular pyramid because the triangles have a common vertex but it is not a pyramid because a base is missing.

A triangular pyramid solid has 4 faces.


A discussion question: Can you count the number of vertices of the envelope? Explain.

Although the pupils only have the envelope of the triangular pyramid they will be able to discover the number of edges,vertices and faces of the triangular pyramid solid.


Answer: The envelope has 4 vertices.


A discussion question: Can you count the number of edges of the envelope? Explain.


Answer: The envelope has 6 edges.


A discussion question: Can you count the number of faces of the envelope? Explain.


Answer: The envelope has 3 faces but the solid has 4 faces as shown in the animation.

The difference in the number of faces must be shown and also another triangular pyramid solid should be shown.


A discussion question: Do you know other pyramids?


Answer: There are square based pyramids,pentagonal based pyramids and hexagonal based pyramid. The pyramid is named after the shape of its base.

There are other pyramids which can be mentioned not all of them being straight pyramids.



A discussion question: In the picture you can see a type of construction looking like a pyramid which was built by the Mayans. Is it a pyramid? Explain.


Answer: It is called a pyramid in Mexico but it is not a geometrical pyramid.


A discussion question: Which of the pictures can be described as a geometrical pyramid?


Answer: Both look like pyramids but only the one marked B has the shape of a geometrical square pyramid because the envelope of the pyramid is made up of triangles.

Only B is a real squared pyramid.


A picture of pupils making their pyramids.
 Lesson Aims
Investigate pyramids
 Description of a pyramid, number of faces,edges and vertices.
 Recognition of a triangular pyramid and envelope of a pyramid.
 Pyramids with different bases.
 Model Name
Envelope of a triangular pyramid.
 Lesson structure
First lesson on the topic of 3D solids.
 Lesson content
Pupils will construct a triangular pyramid, investigate the properties of this pyramid, the envelope of this pyramid constructed from triangles and comparison of the envelope and the solid.
 Prior knowledge
Polygons and triangles.
The envelope of a triangular pyramid