Lesson 10: Solids.Surface Areas (I)May 16, 2012
Three-D shapes have 3-dimensions- length, width and depth. We are going to study some of them: polyhedrons and revolution solids.
A polyhedron is a three-dimensional region of the space bounded by polygons.
Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren’t polyhedrons).
If you click on the links you will learn more about these 3-D shapes in the web where it is taken from the table below, www.mathisfun.com. You can also find questions with answers.
(they must have flat faces)
(if any surface is not flat)
Elements of a polyhedron
Faces: These polygons that limit the polyhedrons.
Edges: Line segments where two faces of a polyhedron meet. They are sides of the faces.
Vertices of the polyhedron: Points at which three or more polyhedron edges of a polyhedron meet.
Diagonals of a polyhedron: Segments, joining two vertices, which are not placed on the same face, are called diagonals of polyhedron. The tetrahedron has no diagonals.
Dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. All dihedral angles between the edges are ≤ 180º.
Polyhedral angle: is the portion of space limited by tree or more faces which meet at a vertex. All polyhedral angles between the edges are ≤ 360º .
Net: It is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. There are several possibilities for a net of a polyhedron.
You can find the net of the main polyhedron and their different possibilities on
2. Regular polyhedrons
2.1. Types of polyhedrons
A convex polyhedron is defined as follows: no line segment joining two of its points contains a point belonging to its exterior. There are many examples known by you: the cube, prisms, pyramids….
A concave polyhedron, on the other hand, will have line segments that join two of its points with all but the two points lying in its exterior.
Below is an example of a concave polyhedron.
The study of polyhedrons was a popular study item in Greek geometry even before the time of Plato (427 – 347 B.C.E.) In 1640, Rene Descartes, a French philosopher, mathematician, and scientist, observed the following formula. In 1752, Leonhard Euler, a Swiss mathematician, rediscovered and used it.
In a convex polyhedron:
If F = number of faces , V = number of vertices and E = number of edges, then
F + V = E +2
This formula is named Euler’s Formula.
All the convex polyhedrons verify this formula. There are some concave polyhedrons that verify it. However, there are concave polyhedrons that don’t verify this formula.
2.2. Regular polyhedrons
A regular polyhedron is a polyhedron where:
- each face is the same regular polygon
- the same number of faces (polygons) meet at each vertex (corner)
They are also called Platonic solids.
There are five Platonic solids:
|1. Tetrahedron, which has three equilateral triangles at each corner.|
|2. Cube, which has three squares at each corner.|
|3. Octahedron, which has four equilateral triangles at each corner.|
|4. dodecahedron, which has three regular pentagons at each corner.|
|5. Icosahedron, which has five equilateral triangles at each corner.|
Exercise: Fill in this table and check they satisfy Euler’s formula. Why?
faces edges vertices tetrahedron ___ ___ ___ cube ___ ___ ___ octahedron ___ ___ ___ dodecahedron ___ ___ ___ icosahedron ___ ___ ___