§ 8.1
1. Prove that the sum of the interior angles of a polygon
of n sides is 180 (n – 2).
P
F
A
E
B
C
D
Note that a polygon of n sides may be dissected into n triangles. The sum
of the angles of these n triangles will be 180n. This sum includes all the
interior angles plus the angles around the point P. hence we need to
subtract that measure of 360 from the total.
180n – 360 = 180 ( n – 2)
2. Prove that in a regular polygon of n sides each interior angle has a measure
of (180(n – 2))/n
A
B
P
E
D
C
From problem 1 we know the sum of the interior angles is 180 (n – 2). And
since this is a regular polygon to get the measure of each interior angle we
need only divide by n the number of interior angles. Hence each interior angle
has a measure of (180 (n – 2))/n. the exterior angle is the supplement of this
3. Prove that in a regular polygon of n sides each exterior angle has a measure
of 360/n
A
B
P
E
D
C
From problem 1 we know the sum of the interior angles is 180 (n – 2). And
since this is a regular polygon to get the measure of each interior angle we
need only divide by n the number of interior angles. Hence each interior angle
has a measure of (180 (n – 2))/n. The exterior angle is the supplement of this
angle and hence has a measure of 180 - (180 (n – 2))/n = 360/n
4. Prove that the sum of the exterior angles of a polygon is 360.
Each interior angle and its adjacent exterior angle are supplementary. For
each vertex the sum of the exterior and interior angles will be 180 for a total
sum of 180n. We know that the sum of the interior angles is 180 (n – 2).
The sum of the exterior angles will then be 180 n - 180 (n – 2) = 360
Note this theorem is for all polygons, not just regular polygons..
5. Make a drawing of each of the eight semiregular tessellations
6. Draw an example of a 2-uniform tiling. Don’t use any of those given in class.
7. Will the figure to the right, a bisected regular hexagon, tessellate the plane?
Even though each shape is a pentagon, together they
form a regular hexagon which tessellates the plane.
8. Research Penrose tiling.
9. Will the figure to the right, a bisected regular hexagon, tessellate the plane?
Even though each shape is a pentagon, together they
form a regular hexagon which tessellates the plane.