Download Polygons are closed, many-sided figures with sides made of

Properties of Polygons
Polygons are closed, many-sided figures with sides made of segments joined only at their endpoints.
The name literally means many angles as poly means many, gon means angle. The polygon with the
least amount of sides is the triangle. Then in order of number of sides the list goes: tetragon, pentagon,
hexagon, heptagon, octagon, nonagon (enneagon), decagon, hendecagon, and dodecagon, etc.
Basically the prefixes to gon are all Greek words for numbers. If you want to know more, do an
internet search and you’ll come up with more than enough names to suit your fancy.
According to the Triangle Sum Theorem, the sum of the angle measures in any triangle will add up
to 1800. To find the sum of the angles of any polygon take the number of sides, subtract 2, and
multiply by 180. Below is an explanation of why this works.
(Number of Sides – 2) x 180 = Sum of the measures of angles in a polygon
a1 + b1 + c1 = 1800
(a1 + b2 ) + c1 + (b1 + a2) + c2 = 3600
a1
a1 b2
c1
b1
c2
c1
b1
a2
The sum of the measure of the angles in each triangle is 1800. A quadrilateral (tetragon) can be
divided into two triangles. The sum of the angle measures of each is 1800; 1800 + 1800 = 3600
Pentagon (5 sides)
5400
Hexagon (six sides)
7200
1800
1800
1800
1800
1800
1800
1800
With each added side, the polygon can be divided into one more triangle. Since each triangle’s sum
of angle measures is 1800, an additional side means another 1800 of angles is formed. For example a
heptagon (7 sides) can be divided into 5 triangles; 5 x 1800 = 9000. Using the formula it would look
like this.
(7 sides – 2) x 1800  (5) x 1800 = 9000
If the polygon is a regular polygon, meaning all the sides and angles are the same measure, then the
measure of each angle can be determined by dividing the total measure of the angles by the number of
angles in the polygon. For example: a hexagon has 6 sides. The sum of the measure of all of its angles
is (6-2) x 180 = 7200. To find the measure of each of its angles divide 7200 by 6. The measure of each
angle would be 1200.
1200
Total of angle measures =
7200
1200
1200
1200
1200
1200
7200  (6 angles) = 1200 per angle
Properties of Polygons
Determine the sum of the measures of the angles in each of the following polygons.
Also for each determine the measure of each individual angle formed by two adjacent sides.
1. A regular tetragon (square)
2. A regular triangle (equilateral triangle)
3. A regular octagon
4. A regular pentagon
5. A regular heptagon
6. A regular nonagon (nine angles)
7. A regular decagon (10 angles)
8. A regular hexagon
9. The sum of the measures of the angles in a regular polygon is 23400. How many sides does it have?
(Use the formula)
10. The measure of one angle of a regular polygon is 1350. How many sides does it have?
Show how you can determine this using simple algebra. (Use the formula)
11. If the shaded polygon below was removed. What regular n-gon would be required to exactly fill
the remaining space?
Hint: Remember the sum of the
central angles that form a circle
is 3600. a + b + c = 3600
a
b c