Folding Polygons From a Circle

1. Folding Polygons From a Circle A circle cut from a regular sheet of typing paper is a marvelous manipulative for the mathematics classroom. Instead of placing an emphasis on manipulating expressions and practicing algorithms, it provides a hands-on approach fro the visual and tactile learner.

2. 1. Mark the center of your circular disk with a pencil. Fold the circle in half. What is the creased line across the disk called? Fold in half again to determine the true center. What are these two new segments called? What angle have you formed? Unfold the circle. How many degrees are there in a circle? Was your estimate of the center of the circle a good one? Compare your result with that of your neighbor.

3. Vocabulary • plane • circle • arc of a circle • degrees in a circle • semicircle • degrees in a semicircle • center of circle • diameter • endpoint • line segment • midpoint of a line segment • radius

4. 2. Place a point on the circumference of the circle. Fold the point to the center. What is this new segment called?

5. Vocabulary • circumference of a circle • area of a circle • chord

6. 3. Fold again to the center, using one endpoint of the chord as an endpoint for your new chord.

7. Vocabulary • sector of a circle

8. 4. Fold the remaining arc to the center. What have you formed? Compare your equilateral triangle with that of your neighbor. Throughout of the rest of this activity suppose that the area of your triangle is one unit.

9. Vocabulary • area of a triangle = 1/2 base x height • triangle • equilateral triangle • isosceles triangle • equiangular triangle • sum of the measures of the angles in a triangle = 180 degrees • base • vertex • point • altitude • mediancircumcenter • incenter

10. Vocabulary • orthocenter • centroid • angle bisector • perpendicular bisector • perimeter of a triangle • scalene triangle • right triangle • hypotenuse • legs of a right triangle • special 30-60-90 degree triangle • Pythagorean theorem • triangle inscribed in a circle

11. 5. Find the midpoint of one of the sides of your triangle. Fold the opposite vertex to the midpoint. What have you formed? What is the area of the isosceles trapezoid if the area of the original triangle is one unit?

12. Vocabulary • trapezoid • parallel vs not parallel sides • isosceles trapezoid • area of a trapezoid = 1/2 height (top base + bottom base) • quadrilateral • fractions • rectangle • right angle • area of a rectangle = length x width • perimeter of a rectangle

13. 6. Notice that the trapezoid consists of three congruent triangles. Fold one of these triangles over the top of the middle triangle. What have you formed? What is its area?

14. Vocabulary • parallelogram • parallel lines • area of a parallelogram • polygon • regular polygon • perimeter of any polygon • rhombus • area of a rhombus • length

15. 7. Fold the remaining triangle over the top of the other two. What shape do you now have? What is its area? The triangle is similar to the unit triangle we started with.

16. Vocabulary • similar • congruent

17. 8. Place the three folded over triangles in the palm of your hand and open it up to form a three dimensional figure. What new shape have you made? What is its surface area?

18. Vocabulary • pyramid • surface area • faces • base • edge

19. 9. Open it back up to the large equilateral triangle you first made. Fold each of the vertices to the center of the circle. What have you formed? What is its area?

20. Vocabulary • hexagon • pentagon • central angles of polygons • sum of the measures of the interior angles of a polygon

21. 10. Turn the hexagon over and with a crayon, pen, or pencil shade the hexagon. Remember what the area of this hexagon is when compared to the original equilateral triangle. Turn the figure over again. Push gently toward the center so that the hexagon folds up to form a truncated tetrahedron. What is its surface area?

22. Vocabulary • tetrahedron • platonic solid • truncated tetrahedron

23. 11. Using only the fold lines already determined, create different polygonal figures and determine their area. Using only the existing fold lines, can you construct figures with the following areas? Are there any others? If so, sketch them. 1 , 1 , 19 , 2 , 3 , 7 , 8 , 7 , 23 4 2 36 3 4 9 9 18 36

24. Vocabulary • fractions

25. 12. You can tape twenty truncated tetrahedra together to make an icosahedron. There will be five on the top, five on the bottom, and ten around the middle.

26. Vocabulary • icosahedron Other vocabulary that might be used: • common denominator • arithmetic of fractions • closed set • bounded set • compact set • interior of a set • quadrants • secant line • Euler Line