Angles in Polygons and Tessellations
Topics in this lesson
- Sum of interior angles in a polygon
- Sum of angles at a vertex of a tessellation
- Vertex type of a tessellation
Theorem: The sum of angles in any plane triangle is 180o.
Here (on the web or activity in class) is a hands-on demonstration of this fact.
A demonstration like this is not a mathematical proof because
(a) it can only show that the angle sum is pretty close to
180o, since real-world measurements are not infinitely precise
and (b) the demonstration considers only the specific triangle
you made, not every triangle that could possibly exist.
The demostration could be made into a proof; see the end of that page
for suggestions.
Pattern blocks
Pattern blocks are widely available math manipulatives; they are flat
pieces made of plastic or wood in 6 different shapes: equilateral triangle,
regular hexagon, square, a trapezoid that is half of a regular hexagon,
and a fat and a thin rhombus. One angle on the fat rhombus is the same
as an angle of the equilateral triangle; one angle on the thin rhombus
has half the measure of an angle of the equilateral triangle. All edges
are equal, except for the long edge of the trapezoid.
You may want to make a pattern block template out of a piece of cardboard. Use poster
board, a cereal box, or the back of a pad of paper; use a utility knife for cutting.
Tessellations by polygons
Recall that a tessellation of the plane by polygons is a collection
of polygons that cover the plane without gaps or overlaps. The polygons
are not necessarily congruent, and they do not necessarily make a repeating
pattern. Here are two special types of tessellations.
Definition: An edge-to-edge tessellation is one
in which edges of adjoining polygons have the same endpoints; that is,
one edge can't end in the middle of another edge. Here is part of a tessellation
that is not edge-to-edge.
All the tessellations in the first two lessons
(Equilateral Triangles
and Constructing Perpendiculars) are edge-to-edge.
Definition: In an edge-to-edge tessellation
where all the polygons are regular,
the type of a vertex is a list of all the polygons around the vertex
in a cyclic order.
For example, in the checkerboard (square graph paper) tessellation,
as you walk in a circle around a vertex, you encounter
square, square, square, square before you get back to where you started.
Usually a regular n-gon is just abbreviated by the number
of sides n. So the type of each vertex in a checkerboard is 4.4.4.4.
The
type of each vertex in the equilateral triangle tessellation
in the first
lesson is 3.3.3.3.3.3.
Definition: A semiregular tessellation is
an edge-to-edge tessellation
where all the polygons are regular,
and all the vertices have the same type.
Group problems
- Find a formula for the sum of the interior angles in an n-gon (a polygon
with n sides, where n is a positive whole number).
- Find a formula for the measure of one angle in a regular n-gon. (All
angles in a regular n-gon are equal.)
- In an edge-to-edge tessellation, what can you say about the
angles of the polygons that meet at each vertex?
- In this problem, answer the questions and draw the tessellations that exist
by tracing pattern blocks.
Note that, in constructing semiregular tessellations,
whenever you add a new piece,
all the vertices of that piece must have the same type, so that
affects the next pieces you add.
- Is there a semiregular tessellation where every vertex has type 6.6.6?
- Is there a semiregular tessellation where every vertex has type 3.3.3.3.6?
- Is there a semiregular tessellation where every vertex has type 3.6.3.6?
- Find all other semiregular tessellations that use equilateral triangles,
regular hexagons, or both, and prove that you have found them all.
- There is a semiregular tessellation in which some
of the polygons are regular octagons.
- What are the other polygons? (Hint: Use your formulas for angles
in regular n-gons.)
- Draw the tessellation.
- Prove that this is the only tessellation with these rules that uses
octagons.
Individual problems
- Suppose you have a tessellation of the plane by polygons which is not
edge-to-edge. What can you say about the angles at such a vertex?
- Find all semiregular tessellations that use both
squares and equilateral triangles.
Prove that you have found them all.
Use your template to draw the tessellations.
- Any triangle can be used to tessellate the plane (copies of only that
triangle). Find an explicit procedure to do this
that will work for any triangle and explain it.
Also give an example (draw the tessellation) with a non-special triangle (not isosceles,
equilateral, or right).
- (Extension) Does the formula for the sum of
interior angles in an n-gon apply to non-convex polygons?
Why or why not?
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