matematicas visuales visual math

A pyramid is a polyhedron with a polygonal face (known as the base) and the other faces are triangles meeting at a common point (the apex of the pyramid). These faces (lateral faces) are triangles.

One particular case is when the base is inscribed in a circle. In the first mathlet we can play with pyramids that have a regular polygon as a base. If the apex is above the center of this circle, then we can say that the pyramid is a right pyramid. A regular pyramid is a right pyramid whose base is a regular polygon.

The main interest of this page is to see how a pyramid can be developed into a plane.

Pyramid and Pyramidal frustum: a pentagonal pyramid | matematicasVisuales
Pyramid and Pyramidal frustum: a pyramid developing| matematicasVisuales

Plane development or net of a pentagonal pyramid:

Pyramid and Pyramidal frustum: plane development or net of a pentagonal pyramid | matematicasVisuales

Another example, the net of an hexagonal pyramid:

Pyramid and Pyramidal frustum: plane development or net of an hexagonal pyramid | matematicasVisuales

To calculate the lateral surface area of a pyramid we need the slant height. The slant height of a pyramid if the distance between the apex and the center of a side of the base. It is the altitude of a lateral face. There is a relation between the slant height and the height of a pyramid (Pythagorean theorem).

Pyramid and Pyramidal frustum: slant height and height of a pyramid | matematicasVisuales
Pyramid and Pyramidal frustum: slant height and height of a pyramid. Pythagorean theorem | matematicasVisuales

We are going to calculate the lateral surface area of a pyramid. If P is the base perimeter, the formula for the lateral surface area of a pyramid (lateral faces are triangles) is like the formula for the area of a triangle:

Pyramid and Pyramidal frustum: lateral surface of a pyramid | matematicasVisuales

When we study the lateral surface area of a cone, the formula will be similar (as Kepler and the area of a circle.)

The most regular pyramid is a tetrahedron.It is a platonic solid made by four equilateral triangles. Then a tetrahedron is a particular case of a triangular pyramid.

Pyramid and Pyramidal frustum: a tetrahedron | matematicasVisuales

And this is the plane net of a tetrahedron:

Pyramid and Pyramidal frustum: plane net of a tetrahedron | matematicasVisuales

When we cut a pyramid by a plane parallel to the base we get a pyramidal frustum (or a truncated pyramid).

For example, this is an hexagonal frustum:

Pyramid and Pyramidal frustum: hexagonal frustum | matematicasVisuales

And this is its plane net:

Pyramid and Pyramidal frustum: plane net of an hexagonal frustum | matematicasVisuales

Another example:

Pyramid and Pyramidal frustum: heptagonal frustum | matematicasVisuales
Pyramid and Pyramidal frustum: plane net of an heptagonal frustum | matematicasVisuales

As before, we need the slant height to calculate the lateral surface area of a frustum:

Pyramid and Pyramidal frustum: slant height of a frustum| matematicasVisuales
Pyramid and Pyramidal frustum: slant height of a frustum and height. Pythagorean theorem | matematicasVisuales

If P is the bottom base perimeter and p the top base perimeter, the formula for the lateral surface area is like the formula for the area of a trapezoid (lateral faces are congruent trapezoids):

Pyramid and Pyramidal frustum: plane net of a truncated pyramid with its slant height | matematicasVisuales

In the examples above bases were regular polygons. But we can consider pyramids whose bases are not regular polygons. In the next mathlet, bases are non-regular polygons (although they are inscribed in a circle and they are convex polygons). Each time we change the number of sides of the base a new prism is generated with sides randomly drawn:

MORE LINKS

Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (2): Prisms cut by an oblique plane
Plane nets of prisms with a regular base with different side number cut by an oblique plane.
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Plane developments of geometric bodies: Octahedron
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Volume of an octahedron
The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Chamfered Cube
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.