2.6

How Can Build It?

Pg. 19

Pinwheels and Polygons

2.6 – How Can I Build It?______________Pinwheels and Polygons

In this section you will discover the names of the many different polygons and how they are classified.

2.30 – PINWHEELS AND POLYGONSItzel loves pinwheels. One day in class, she noticed that if she put three congruent triangles together, that one set of the corresponding angles are adjacent, she could make a shape that looks like a pinwheel.

a. Can you determine any of the angles of her triangles? Explain how you found your answer.

3603 120°

120°120°

b. The overall shape (outline) of Itzel's pinwheel is shown at right. How many sides does it have? What is another name for this shape?

6 sides

120°120°120°

1

2

3

45

6hexagon

c. Itzel's shape is an example of a polygon because it is a closed, two dimensional figure made of straight line segments connected end-to-end. As you study polygons in this course, it is useful to use these names because they identify how many sides a particular polygon has. Some of these words may be familiar, while others may be new. Fill in the names of the polygons below. Then, draw an example of a heptagon.

Name of Polygon # of sides

3

4

5

6

7

triangle

quadrilateral

pentagon

hexagon

heptagon

Name of Polygon # of sides

8

9

10

12

n

octagon

nonagon

decagon

dodecagon

n – gon

Then, draw an example of a heptagon.

2.31 – MAKING PINWHEELSItzel is very excited. She wants to know if you can build a pinwheel using any angle of her triangle. Obtain a set of triangles from your teacher. Work with your team to build pinwheels and polygons by placing different corresponding angles together at the center. You will need to use the triangles from all four team members together to build one shape. Be ready to share your results with the class.

3 triangles120°

11

1

Angle ## of triangles

neededMeasure of the central angle

1

2

3

3 120°

9 triangles40°

Angle ## of triangles

neededMeasure of the central angle

1

2

3

3 120°

9 40°

18 triangles20°

Angle ## of triangles

neededMeasure of the central angle

1

2

3

3 120°

9 40°

18 20°

2.32 –PINWHEEL PATTERNSJorge likes Itzel's pinwheels but wonders, "Will all triangles build a pinwheel or a polygon?”

a. Use the different triangles provided by your teacher. Work together to determine which congruent triangles can build a pinwheel (or polygon) when corresponding angles are placed together at the center. If it works, fill in the table.

Angle ## of triangles

neededMeasure of the central angle

A

B

C

D

E

F

8 45°

Not possible

5 72°

Not possible

12 30°

Not possible

b. Explain why one triangle may be able to create a pinwheel or polygon while another triangle cannot.

It must divide by 360° evenly

c. Jorge has a triangle with interior angle measures 32°, 40°, and 108°. Will this triangle be able to form a pinwheel? Explain? If so, at what angle?

Yes, the 40° divides in evenly

2.33 –POLYGONSJasmine wants to create a pinwheel with equilateral triangles.

a. How many equilateral triangles will she need? Explain how you know.

60°

36060

660°60°

60°60°

60°

b. What is the name for the polygon she created?

hexagon

60°60°60°

60°60°

60°

c. Jasmine's shape is an example of a convex polygon, while Inez's shape, shown at right is non-convex (or concave). Study the examples below and write a definition of a convex polygon on your paper.

Has vertices going inward, like a cave

All vertices point outward

2.34 –CONCAVE VS. CONVEXBrenda noticed that the non-convex (concave) shapes all had a part that went inward, like a cave. She decided to investigate more. Sort the shapes below as either "convex" or "concave".

convex concave

convex concave

2.35 –EQULATERAL, EQUIANGLUAR, AND REGULAR

Brenda was curious about the relationship between the sides and angles of polygons. When all sides are equal, it is called equilateral. When all angles are equal, the polygon is called equiangular. When it has all equal sides AND all equal angles it is called regular.

Classify the name of the polygon by the number of sides. Is the polygon equilateral, equiangular, or regular? Then determine if it is convex or concave.

Name: _______________

Equilateral, Equiangular, Regular

Convex OR Concave

hexagon

Name: _______________

Equilateral, Equiangular, Regular

Convex OR Concave

pentagon

Name: _______________

Equilateral, Equiangular, Regular

Convex OR Concave

octagon

Name: _______________

Equilateral, Equiangular, Regular

Convex OR Concave

decagon

Name: _______________

Equilateral, Equiangular, Regular

Convex OR Concave

heptagon

Name: _______________

Equilateral, Equiangular, Regular

Convex OR Concave

quadrilateral