Polyhedra Constructions

·**Constructions** for Polyhedron Project:Equilateral Triangle:1. Draw a line2. Measure off length you want (4"?)3. Put your compass point on one end of this length and your pencil point on the other.4. Make an arc above the line5. Rip your compass over, and with the same setting, make another arc above the line. The arcs should intersect.6. Connect each endpoint of the line to intersection of these arcs. The result should be an equilateral triangle.Square:1. Draw a line. .2. Measure off the length you want on the line.3. Extend the line so it goes at least an inch further on both ends.4. Put your compass point at one end of the measured length. .5. Open your compass to any seqing (about an inch would be good) and make an arc on either side of thisendpoint6. These arcs intersect the line throught the measured length. Label the points of intersection A and B.7. Put your compass on A and open your compass more than half way to B.8. Make arcs above and below the line.9. Rip your compass over and with the point on B, make arcs above and below the liqe.10. The arcs you drew from A and B should intersect above and below. Label these points of intersection Cand D. Draw a line connecting C and D, and extending in both directions past C and D. The line should beperpendicular to the original length and should pass through one endpoint of the desired length (4" linesegment)11. Repeat steps 5-10 of this construction at the other endpoint of the desired length.12. Now put your compass point on one end of your original desired length and open it until you pencil pointreaches the other endpoint13. Use this compass setting to make an arc above the line. continue the arc until it intersects the perpendicular.14. Rip your compass over and repeat at this other end point .15. You should now have a point of intersection on each of your arcs. Connect these points of intersection.The result should now be a square.Regular Hexagon:I. Measure off your desired length with your compass.2. Draw a circle with this radi us.3. Put your compass point on the circle and make an arc that intersects the circle.4. Move your compass point to this point of intersection and make another one.5. Repeat until you get back to where you started. If you did not end up EXACfLY where you started start overwith step 3.6. Connect the points of intersection on the circle. The result should be a regular hexagon.Regular pentagon with fixed side length.1. Draw a line.2. Measure off the length you want (at least 4") on the line. Label the end points of your desired length A andB. .3. Make sure your line extends at least an inch further than A and B.4. At A construct. a perpendicular line through A. (steps 5 through 10 from the square construction)5. Extend this perpendicular really far (more than twice the length of AB.)6. Put your compass point on A open it until the pencil point is on the other end point, B.7. With the compass point on A, make an arc above the line so that it intersects the perpendicular. Label thispoint of intersection D. Move you compass point to D and make another arc above the point intersection on theperpedicular. Where this second arc intersects the perpendicular, label it C. The result should be that you'vemarked off a distance twice as long as your original desired length.8. Connect B to C.9. On a second sheet of paper (or 2 taped together), draw a long (no, longer) line. Label one endpoint A.10. Copy AB onto the line. From the end of this copied segment. copy CB onto the line. The result should bethe segments AB and BC next to each other forming a long line. '"

11. Bisect this long segment line. AC. [your compass may not be big enough to do this bisection, You mightwant to do it with folding. Fold one end point onto the other endpoint. Crease the fold. When you UP -( 'I( th,'paper. you should notice that the crease crossed the line. This is the midpoint. Label the midpoint h lOB'segment. M.12. On a third sheet of paper. draw a line length though the middle of the page. lengthwise. Copy AM onto thisline.13. Put your compass point on A and open it until the pencil reaches M.14. From A, make a long arc above the line with AM on it.15. Go back and measure off AB with your compass.16. Put your compass point on M (with the AB length still on your compass) and make an arc above the lineuntil it intersects the arc you drew in step 16. Label this point of intersection X. From point M also make an arcbelow AM.17. Put your compass, still set with the AB length. on A. Make an arc below and above AM .18. Put your compass. still set with the AB length. on X. Make an arc that intersects the one you made aboveAM from A in step 19. Call this poiot of intersection Y.19. Notice that the arcs you made below AM from A and M now intersect. Call this point of intersection Z.20. Connect A to Z. Z to M, M to X. X to Y, and Y to A.21. The big test: To see if it really worked you should bisect any two of the sides you created in step 22. Putyour compass point on the point where these segment bisectors intersect. This should be the center ofthepentagon. just like we used chord bisectors to find·the center ofa circle. Open your compass from this centerpoint to A. Make a big circle...as you make it, the pencil line should pass through all the points A, Z,M, X, and Y. If it doesn't...you don't have a regular pentagon. Go back and fwd your mistake.Octagon comes from the square construction. Decagon comes from the Pentagon construction andDodecagon comes from the hexagon construction. See directions at bottom of page to determine theappropriate side length to start with so you end up with the side length you want.All three of these constructions are done in the same manner. Let's show how the square turns into an octagonas an example:1. With your square, draw in the diagonals. They should intersect in the center of your square.2. Put your compass point on the center of the square. Open the compass until the pencil reaches the corner.3. Make a circle with this radius. using the same center (i.e. don't lift your compass before making the circle).Your circle should pass through aU the comers of the square** IF your circle does NOT pass through all thevertices then one of two things has happened. Either you made a mistake with the original polygon constructionor you made a mistake finding the center of the polygon (these last few steps.) Regardless, what you've justlearned is that your polygon is NOT regular and you MUST redo it. Sorry. Be patient.4. Bisect each of the sides of the square. The bisectors should pass through the circle at 4 points. Label thesepoints of intersection.5. Connect the original corners with these new bisector points (step 4) along the circumference of the circle.The result should be a regular octagon.To get decagon or dodecagon, bisect 2 different sides of your polygon. These 2 bisectors should intersectinside the polygon. Put your compass point on this point of intersection. Open your compass to reach one ofthe vertices of your polygon. Draw a circle with this radius without lifting your compass. This circle shouldpass through all the vertices of your polygon.** Bisect all the sides of your polygon. Label the points wherethese bisectors intersect the circle. Connect these points with the original vertices of the polygon along thecircumference. If you started with pentagon you now have a decagon. If you started with hexagon, you nowhave a dodecagon.But you might notice that the sides of your new octagon, decagon, or dodecagon would be too small.So use the following formula to help figure out what side length to use on your original constructionso that when you bisect the sides you end up with the proper lengths you want:x = 2L. Sin((90n-I80) + n)where x=the length you should start your construction with, L= the length you want to end up withn=the number of sides you want to end up with. So for example, to get 4" sides make a pentagon with 7.6"sides and a hexagon with 7 9.7/16" to get 4" sides.