Squares, Rectangles, Parallelograms and Other Polygons

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K-12: Materials at high school level.



We will be talking about all kinds of polygons. Below we will talk about and show some examples of the most common ones including triangles, quadrilaterals (4-sided shapes), etc.

Contents

Triangles

Triangles are just shapes with 3 (straight) sides. They can be big or small and can look somewhat different. Depending on the angles and the sides we can sort the triangles into different types.

Triangles.svg

The simplest way to sort triangles is by their angle size:

  • Acute triangle: An acute triangle is one in which all the angles are acute (less than 90°).
  • Obtuse triangle: An obtuse triangle is one in which one of the angles is obtuse (more than 90°).
  • Right triangle: A right triangle is one in which one of the angles is a right angle (exactly 90°).

It is not hard to see that every triangle falls into exactly one of these three groups. Every triangle is either an acute triangle, an obtuse triangle, or a right triangle.


Another way to group triangles is by looking at the lengths of their sides:

  • Equilateral triangle: An equilateral triangle is one in which all three sides have the same length.
  • Isoceles triangle: An isoceles triangle is one in which two sides have the same length.
  • Scalene triangle: A scalene triangle is one in which all three sides have different lengths.

Note that every equilateral triangle is automatically also isosceles.

4-sided polygons (Quadrilaterals)

After triangles, the type of shape we will encounter the most is the quadrilateral:

  • Quadrilateral : A polygon with four sides.
Quadrilaterals.svg

Note that in general the 4-gons can look pretty strange and irregular. There are a handful of special cases we care about. (See below)

Squares

  • Square: A quadrilateral with four equal-length sides and four right angles.
Squares.svg

Note that the square can be of differnt sizes and you are allowed to rotate it. Above are several examples of squares.

Rectangles

  • Rectangle: A quadrilateral having four right angles.
Rectangles.svg

Again, the rectangles come in different sizes and we can draw them at an angle if we want. As long as all the angles are 90 degrees, it will be a reactangle.

Parallelograms

  • Parallelogram: A quadrilateral with two pairs of parallel sides.

We haven't technically defined what parallel means, but it means that the two segments never meet, even if you were to continue tem infinitely far. Known examples are for instance the left and the right side of a ladder, or the double lines you see on some freeways.

Parallelograms.svg


Rhombuses

Rhombus
A quadrilateral having all four sides of equal length.
Rhombuses.svg

If you have played cards this is the shape of diamonds :)


Polygons in General

Polygon: A polygon is a closed planar figure made by joining line segments. The segments may not cross, and each segment must connect to exactly two others at its endpoints.

Below are some examples of things that are polygons and things that are not polygons.

Polygons
Polygons.svg
Not Polygons
Notpolygons.svg

The left figure is not closed, and the figures in the middle are not made of line segments. The figure on the right is not a polygon, since its sides intersect each other.

  • Vertex: A vertex of a polygon is a point where two sides come together.


The word "vertex" is more precise than the common term "corner", because "corner" has many other uses in English. The plural of "vertex" is "vertices" - a triangle has three vertices.

The way we identify a polygon is usually by the number of sides it possesses, which is the same as its number of angles.

A 28-gon

Classifying polygons by number of sides is important enough that there are special words for polygons with small numbers of sides:

# of sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Enneagon
10 Decagon


There are (ridiculous) names for polygons with many more sides (see wikipedia:Polygon), but generally for larger numbers of sides, one uses the number of sides followed by "-gon". We talk about 7-gons and 8-gons for instance (instead of the harder to remember names heptagon or octagon).

Worksheet