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.

Plane development or net of a pentagonal pyramid:

Another example, the net of an hexagonal pyramid:

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).

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:

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.

And this is the plane net of a tetrahedron:

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:

And this is its plane net:

Another example:

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

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):

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:

LINKS

	Plane developments of geometric bodies (6): Pyramids cut by an oblique plane Plane net of pyramids cut by an oblique plane.
	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 (4): Cylinders cut by an oblique plane We study different cylinders cut by an oblique plane. The section that we get 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.
	The volume of the tetrahedron The volume of a tetrahedron is one third of the prism that contains it.
	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 One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.