Shapes


Polygons - Introduction


The word polygon is a general term describing a shape.

The names given to polygons with specific numbers of sides include

No. of sides Name of Polygon
3 Triangle
4 Quadrilatreal
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon

Regular Polygons are polygons with all sides of equal length.

All their angles will also be equal, although this feature does not necessarily apply in the inverse direction, i.e. a polygon with all angles equal is not necessarily regular (note rectagles, which have all angles equal but are not regular)

Internal Angles


The formula for the internal angles of a polygon is

S = (n-2) × 180 °

where S is the total sum of the internal angles and n is the number of sides of the polygon

You can check that this works for a triangle (n=3), where the internal angles ewqual 180 °, and for quadrilaterals (n=4), where the internal angles equal 360°.

Quick Quiz    

What size are the internal angles of a regular hexagon?

Quick Quiz (Harder Example)    

A regular polygon has exterior angles of 135°. How many sides does it have?

Exterior Angles

In the adjacent figure, a is an interior angle and b is called the external angle. You can see that the sum of the interior angle and a corresponding external angle is 180°.

We could just as easily extending the line coming into a from a 'vertical' direction and produced an exterior angle below the angle a. This external angle would obviously be of the same size as the exterior angle shown in the figure.

The sum of the exterior angles of a polygon is 360°. Just to clarify this, this sum involves considering only one external angle at each apex, e.g. in the adjacent figure b would be the only exterior angle to a. This concept gives us another method for calculating the interior angles of a regular polygon. For example, a regular polygon will have exterior angles of

360 / 5 = 72°

Since the sum of ther interior and exterior angle is 180, the interior angle of a regular pentagon will be

180 - 72 = 108°

Quick Quiz    

A regular polygon has exterior angles of 30°. How many sides has it?

Similar Figures


Stated baldy, two figures are similar if they have the same shape. They have the same angles but one is an enlargenment of other

To be more specific, each side of one figure will be a constant multiple of the corresponding sides of the other figure. This constant multiple is called the scale factor .

The ratio of two corresponding sides is identical for the two figures.

Example


These two triangles above are similar, and if the scale factor was 1.2 then

DE = 1.2(AB);    EF = 1.2(BC);    DF = 1.2(AC)

and we can state, to give one ratio as an example, that

If we knew three of these values but one was unknown, then we could solve for the unknown by treating this expression as an algebraic equation. In that case it would be best if the unknown was one of the numerators.

Using a bit of algebra on the above expression will give


Another Example


One of the ratios that could be derived from the above scenario would be

Quick Quiz    

a) A stick 2m long is placed vertically so that its top is in line with the top of a tree, from a point A, which is 3m from the stick and 30m from the tree. How tall is the tree?



Congruent Figures


Congruent figures are shapes with corresponding sides of exactly equal length.

Tesselations


The hexagon is the highest order of regular polygon that can be used to tile a floor without leaving gaps.

In the case of beehives, this is an ideal comprommise. The higher the order of a polygon (i.e. the larger number of sides it has), the greater the area it can cover for a given perimeter length (and in this sense a circle can be considered the limiting figure as you keep on increasing the number of sides). However packing circles is not as efficient for a bee as packing hexagons.