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Interior angles of polygons
- Draw a convex polygon with at least four sides. Choose
one vertex, and draw diagonals from that vertex to the
other vertices of the polygon.
To
test
if
a
polygon
is
convex,
imagine
connecting
each
vertex
to
every
other
vertex.
If
all
connecting
segments
stay
inside
the
polygon,
it
is
convex.
- The segments you drew divided the polygon into
several smaller figures. What kind of figures are they?
- What is the sum of the measures of the angles in each
of the figures?
- An interior angle of a polygon is an angle inside the
polygon where two sides meet. Use your answers above
to find the sum of the interior angles of your polygon.
- Explain why the sum of the interior angles is going
to be the same for any convex polygon with the same
number of sides.
- Complete the following table. If necessary, draw an example of
each polygon and use problem 1 to help you.
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|
| | Convex polygon name | Number of sides | Sum of interior angles |
|
|
| | Triangle | 3 | |
| Quadrilateral | 4 | |
| Pentagon | 5 | |
| Hexagon | 6 | |
| Heptagon | 7 | |
| Octagon | 8 | |
| Nonagon | 9 | |
| Decagon | 10 | |
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|
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- Find a pattern in the numbers above. Then describe a quick
way to find the sum of the interior angles of a convex polygon
when you know the number of sides.
- Explain why your rule will work for every polygon.
- A regular polygon is a polygon in which every interior angle
has the same measure and every side has the same length.
What is the measure of an interior angle of
- a regular (equilateral) triangle?
- a regular quadrilateral (square)?
- a regular octagon?
- Describe how to find the measure of an interior angle of a
regular polygon when you know the number of sides.
Answers
-
- Triangles
- Answer depends on the polygon drawn.
- Answer depends on the polygon drawn.
- For a convex polygon with the same number of sides,
there will be the same number of triangles inside. Each
triangle has a sum of 180o, so the sum of all the
triangles’ angle measures--which is also the sum of the
polygon’s interior angles--will be the same.
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|
| | Polygon name | Number of sides | Sum of interior angles |
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|
| | Triangle | 3 | 180 |
| Quadrilateral | 4 | 360 |
| Pentagon | 5 | 540 |
| Hexagon | 6 | 720 |
| Heptagon | 7 | 900 |
| Octagon | 8 | 1080 |
| Nonagon | 9 | 1260 |
| Decagon | 10 | 1440 |
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- Possible answer: Subtract two from the number of sides and
multiply by 180.
- Possible answer: The number of vertices is the same as the
number of sides. When you draw the diagonals, you draw to all
but three of the vertices--the vertex you’re drawing from,
plus the two adjacent ones. One diagonal divides the
polygon into two regions; drawing another adds one to the
number of regions. So you get one more region (triangle)
than the number of diagonals you draw, giving two fewer
triangles than you have sides. Since each triangle has
an angle sum of 180o, you get 180 times two less than
the number of sides for the interior angle sum of the
polygon.
-
- 60o
- 90o
- 135o
- Possible answer: Divide the sum of the interior angles by the
number of sides.
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