Polygons – Polygon basics – In Depth

The word polygon
is a combination of two Greek words: “poly” means many and “gon”
means angle. Along with its angles, a polygon also has sides and vertices.
“Tri” means “three,” so the simplest polygon is called
the triangle, because it has three angles. It also has three sides and three
vertices. A triangle is always coplanar, which is not true of many of the
other polygons.

A regular polygon
is a polygon with all angles and all sides congruent, or equal. Here are some
regular polygons.

We can use a
formula to find the sum of the interior angles of any polygon. In this formula,
the letter n stands for the number of sides, or angles, that the polygon has.

sum
of angles = (n – 2)180°

Let’s use the
formula to find the sum of the interior angles of a triangle. Substitute 3
for n. We find that the sum is 180 degrees. This is an important fact to remember.

sum
of angles = (n – 2)180°

= (3 – 2)180° = (1)180° = 180°

To find the
sum of the interior angles of a quadrilateral, we can use the formula again.
This time, substitute 4 for n. We find that the sum of the interior angles
of a quadrilateral is 360 degrees.

sum
of angles = (n – 2)180°

= (4 – 2)180° = (2)180° = 360°

Polygons can
be separated into triangles by drawing all the diagonals that can be drawn
from one single vertex. Let’s try it with the quadrilateral shown here. From
vertex A, we can draw only one diagonal, to vertex D. A quadrilateral can
therefore be separated into two triangles.

If you look
back at the formula, you’ll see that n – 2 gives the number of triangles
in the polygon, and that number is multiplied by 180, the sum of the measures
of all the interior angles in a triangle. Do you see where the “n –
2” comes from? It gives us the number of triangles in the polygon. How
many triangles do you think a 5-sided polygon will have?

Here’s a pentagon,
a 5-sided polygon. From vertex A we can draw two diagonals which separates
the pentagon into three triangles. We multiply 3 times 180 degrees to find
the sum of all the interior angles of a pentagon, which is 540 degrees.

sum
of angles = (n – 2)180°

= (5 – 2)180° = (3)180° = 540°

Related Links:
Polygon definitions,
p olygon formulas (area, perimeter) and polygon names
(Tables and Formulas)

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