Activity 10.1 Paper Circle Polygons

Math Activity: The Things You Find in a Paper Circle

Purpose: Explore basic geometric concepts and plane figures.

Materials: Paper, scissors, pencil

Directions: below or printable version Download printable version

Find a round object to trace out a circle on a piece of paper. Try to find something with a diameter of 10 to 20 cm. It is is too small the folding will be difficult.

* A circle is formed on a piece of paper by a line that is the same distance from a specific point called the center of the circle.

How can you find the center of a paper circle? We are about to try.

∆ Using a circle cut out of paper, fold the circle in half.
* This shape is called a semi-circle.

1. How much of the area of the circle is in the semi-circle?

∆ Open and then fold a new semi-circle.

∆ Open up the circle.

2. What do you see about the fold for the second semi-circle?

3. What happens when you fold a third semi-circle?

* All folds intersect at the center of the circle.

4. Can you see why?

* Any fold that passes through the center of the circle forms a diameter.

∆ Open up the circle and make a small dot for the center.

∆  Fold the edge of the circle to the center and crease.

* All folds not passing through the center are called chords of the circle.

∆ Make a second fold to the center so that the end of the new chord meets the end of the first chord and the edge of the circle is at the center of the circle.
The figure now resembles an ice cream cone.

∆ Make a third fold to the center so that the end of the new chord meets the ends of the previous two chords.
If all goes well, then the edge of the circle also meets the center of the circle.

5. What shape is this?

* A three-sided figure with sides the same length is called an equilateral triangle.
Open and see the equilateral triangle inscribed in the circle.

∆ With the equilateral triangle, find the midpoint of a chord by folding a side so the adjacent vertices meet. Make a crease in the middle of the chord.

∆ Bring the tip of the opposite vertex to this midpoint.

6. What shape is formed?

7. The area of the trapezoid is what part of the equilateral triangle?

* The line connecting the midpoint of two sides of a triangle is parallel to and equal to one-half of the base.

∆ With the trapezoid, fold a vertex of the original triangle to its’ opposite vertex.

8. What figure is formed?

Let's make some comparisons using fractions.

9. The area of the rhombus is what part of the equilateral triangle?

10. The area of the rhombus is what part of the trapezoid?

* The sides of a rhombus are all equal. Opposite sides are parallel. Opposite angles are equal. A rhombus is a subset of the parallelogram.

∆ With the rhombus, fold the last vertex of the original triangle to its’ opposite vertex.

11. What figure is formed?

Let's make some more comparisons using fractions:

12. The area of the new triangle is what part of the original triangle?

13. The area of the new triangle is what part of the trapezoid?

14. The area of the new triangle is what part of the rhombus?

15. The area of the new triangle is what part of the new triangle?

16. The original triangle is ______ times as large as this new triangle.

17. The trapezoid is ______ times as large as this new triangle.

18. The rhombus is ______ times as large as this new triangle.

19. The original triangle is ______ times as large as the trapezoid and ______ times as large as the rhombus.

∆ Open the figure to the original triangle. Fold the original triangle so the vertices meet.

20. What is this figured called?

* Tetrahedrons have all surface areas congruent.

21. What is the surface area of the tetrahedron if the area of the original triangle is 1?

∆ Open the figure to the original triangle. Fold one vertex to the center of the circle.

22. What new figure is formed?

23. The top base of the trapezoid is ______ the length of the bottom base.

∆ Fold a second vertex to the center of the circle.

24. What is this new figure called?

∆ Fold the third vertex to the center of the circle.

25. What is this figure called?

26. The area of the hexagon is ______ that of the original triangle.

27-29. The area of the hexagon is ______ that of the second trapezoid, ______ that of the pentagon, and ______ that of the small triangle.

∆ Tuck one triangle of the hexagon into a second flap and tuck the remaining flap underneath.

* The new figure is a truncated tetrahedron.

30-34. The surface area of the truncated tetrahedron is:

______ that of the original triangle,

______ that of the smallest triangle,

______ that of the second trapezoid,

______ that of the first trapezoid,

______ that of the rhombus .