3.06 Angles of polygons
State the sum of the interior angles in a quadrilateral.
State the sum of exterior angles for any polygon.
Consider the quadrilateral in the adjacent diagram.
Find the value of the angle marked y.
Find the value of the angle marked x.
Consider the non-convex hexagon in the diagram shown.
Find the least number of triangles that it can be divided into.
Determine the interior angle sum of the non-convex hexagon.
Consider the convex hexagon in the adjacent diagram.
Find the least number of triangles that the hexagon can be divided into.
Determine the interior angle sum of the hexagon.
Consider the adjacent diagram.
The quadrilateral is divided into two triangles (this is the least number of triangles that a quadrilateral can be divided into). The interior angle sum of a quadrilateral is therefore equal to double the interior angle sum of a triangle.
Complete the following table.
| Name of the polygon | Number of sides | Least number of triangles | Interior angle sum |
|---|---|---|---|
| \text{Pentagon} | |||
| \text{Hexagon} | |||
| \text{Heptagon} | |||
| \text{Octagon} | |||
| \text{Nonagon} | |||
| \text{Decagon} |
Consider the pentagon to the right.
Find the value of the following:
a + b + c
p + q + r
x + y + z
Determine the angle sum of the pentagon.
How many non-overlapping triangles can an n-sided figure be split into?
Hence or otherwise, state the interior angle sum of an n-sided polygon.
Find the interior angle sum of a regular polygon whose exterior angles measure 40\degree.
Consider the adjacent quadrilateral.
Find the value of the following angles:
x
a
b
c
d
State the sum of exterior angles in a quadrilateral.
Find the value of the variable(s) in the following diagrams.
Given that each of the following polygon is regular, find the value of x.
Consider the following diagram.
Calculate x.
The interior angles of an n-sided regular polygon each have a measure of x\degree .
Write down a formula for x in terms of n.
Find the measure of each of the interior angles in a regular:
6-sided polygon
Octagon
Decagon
Dodecagon
Find the measure of each exterior angle of a regular polygon with 20 sides.
Consider the regular polygon in the adjacent diagram.
Solve for x.
Solve for y.
Given that the shape in the diagram is a regular pentagon, find the value of x.
Find the value of the variable in the following diagrams.
The exterior angles of an n-sided regular polygon each have a measure of x\degree.
Write down a formula for x.
Given the following angle measures of a regular polygon:
Find the number of sides of the polygon.
Identify the shape of the polygon if it exists.
Interior angles equal to 150 \degree each.
Interior angles equal to 140 \degree each.
Interior angles equal to 120 \degree each.
Exterior angles equal to 36 \degree each.
Exterior angles equal to 72 \degree each.
Exterior angles equal to 45 \degree each.
Exterior angles equal to 75 \degree each.
Exterior angles equal to 50 \degree each.
Interior angles equal to 100 \degree each.
Interior angles equal to 130 \degree each.
Find the measure of an interior angle in each of the following regular polygons:
Hexagon
22-gon
Find the sum of the interior angles in each of the following polygons:
Decagon
22-gon
What is the sum of the exterior angles of a 13-gon?
Solve for x in the following polygons:
Solve for x.
In the diagram, m\angle B= 43 \degree, m\angle FJI= 109\degree, m\angle GHI= 92 \degree, m\angle JFG= 101 \degree, and m\angle C= 45 \degree.
Solve for the following:
m\angle A
m\angle FGH
m\angle HIJ
m\angle BFG+ m\angle EGH + m\angle DHI + m\angle CIJ + m\angle AJF
A house is built with a pitched roof. The engineer states that the angle of the pitch is 14 \degree.
Solve for x.
Determine the number of sides of a polygon when:
The interior angles sum to 360 \degree
Each interior angle has a measure of 135 \degree
Each exterior angle has a measure of 24 \degree
Each exterior angle has a measure of 15 \degree
The interior angles sum to 1980 \degree
Each interior angles has a measure of 160 \degree
ABCDE is a regular pentagon.
What is the measure of \angle AED? Explain how you know.
Explain why x=36 \degree.
Given: Hexagon UVWXYZ
Divide the hexagon into six triangles, all of which share a common vertex T on the interior of UVWXYZ.
Show that the sum of the angles in the six triangles is 1080 \degree.
Show that the sum of the six angles that meet at T is 360 \degree.
Using the results from parts a) and b), show that the sum of the interior angles of Hexagon UVWXYZ is 720 \degree.
The pattern on a soccer ball is a tesselation, or a repeated pattern with no overlaps or gaps. On a three-dimensional soccer ball, the tesselation consists of regular pentagons and hexagons:
When laid flat, a piece of the pattern looks like this:
Explain why the pattern on a soccer ball is impossible to tesselate on a two-dimensional surface.