polyhedra by paper folding worksheet

AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC) AIMING HIGH POLYHEDRA BY PAPER FOLDING A regular polygon has all its angles equal and all edge lengths equal. In a regular polyhedron all the faces are congruent regular polygons and the same number of polygons meet at each vertex. Each regular polyhedron has its own codename. The tetrahedron is 333, the octahedron is 3333, the icosahedron is 33333, the cube is 444, and the dodecahedron is 555. 1. Can you crack the code and explain why these codenames have been given to the polyhedra? 2. Follow the instructions below and make the five regular polyhedral. First fold paper to make regular polygons, then interlock the polyhedra together. The sides of the paper are shown in different colours to make the directions easier to follow. a. Cube The ratio of the short to long edges of your rectangular piece of paper must be 3:4 so remove a 17mm wide strip from a sheet of A4 paper to decrease the length of the longer edge. Fold the paper as shown making 2 folds lengthwise and then 3 folds across the strip. Do the same with three separate sheets of paper, and interlock them to form a cube. b. Tetrahedron Fold an A4 sheet of paper as shown to make a tetrahedron net. Make another identical net, then slot the two nets together to form a rigid tetrahedron. [Note: Use scrap paper. The standard size A4 office paper, used very widely worldwide (but not in the USA) has edge lengths in the ratio √2 to 1. Other sizes are in the same proportion, bigger sizes A1, A2 and A3, and smaller sizes A5, A6 etc. If you do not have A4 paper you can cut your paper in this proportion to make the models.]

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Page 1: POLYHEDRA BY PAPER FOLDING worksheet

AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC)

AIMING HIGH

POLYHEDRA BY PAPER FOLDING Aregularpolygonhasallitsanglesequalandalledgelengthsequal.Inaregularpolyhedronallthefacesarecongruentregularpolygonsandthesamenumberofpolygonsmeetateachvertex.

Eachregularpolyhedronhasitsowncodename.Thetetrahedronis333,theoctahedronis3333,theicosahedronis33333,thecubeis444,andthedodecahedronis555.

1. Canyoucrackthecodeandexplainwhythesecodenameshavebeengiventothepolyhedra?

2. Followtheinstructionsbelowandmakethefiveregularpolyhedral.Firstfoldpapertomakeregularpolygons,theninterlockthepolyhedratogether.Thesidesofthepaperareshownindifferentcolourstomakethedirectionseasiertofollow.

a. CubeTheratiooftheshorttolongedgesofyourrectangularpieceofpapermustbe3:4soremovea17mmwidestripfromasheetofA4papertodecreasethelengthofthelongeredge.Foldthepaperasshownmaking2foldslengthwiseandthen3foldsacrossthestrip.Dothesamewiththreeseparatesheetsofpaper,andinterlockthemtoformacube.

b. Tetrahedron

FoldanA4sheetofpaperasshowntomakeatetrahedronnet.Makeanotheridenticalnet,thenslotthetwonetstogethertoformarigidtetrahedron.

[Note:Usescrappaper.ThestandardsizeA4officepaper,usedverywidelyworldwide(butnotintheUSA)hasedgelengthsintheratio√2 to1.Othersizesareinthesameproportion,biggersizesA1,A2andA3,andsmallersizesA5,A6etc.IfyoudonothaveA4paperyoucancutyourpaperinthisproportiontomakethemodels.]

Page 2: POLYHEDRA BY PAPER FOLDING worksheet

c. OctahedronMakefourtetrahedronnetsfollowingtherecipeabove.Slotthemtogethertoformanoctahedron.Slottwonetstogethertoformwhatlookslikeasquarepyramidwithflapsattached.Dothesamefortheothertwonets.Nowslotthesetwosquarepyramid-likeobjectstogether.

d. DodecahedronIfyoufollowtheseinstructionscarefully,usingpaperwithedgelengthsintheratio√2to1,youwillbeabletomakeadodecahedronliketheoneinthepicture.

e. IcosahedronFromthefinalstageofthetetrahedronrecipe,makethefoldsshownbelowtoproduceatruncatedtetrahedron.Maketwentyofthesetruncatedtetrahedraandgluethemtogethertoformanicosahedron.

Page 3: POLYHEDRA BY PAPER FOLDING worksheet

HELP Sharetheworkinagroupandeachgroupmightmakejustoneofthepolyhedra.Theneachlearneronlyhastomakeafewofthecomponentsthatgotomakingoneofthepolyhedra.Thegroupsshouldlistallthepropertiesoftheirpolyhedron,andthenpassittoanothergroupsothatallthelearnerscanhandleall5polyhedraandcontributetothelistsofproperties.

NEXT IfyouuseA4paperfortheconstructiontomakethedodecahedron,andtrytomakeregularpentagons,thereisasmallerrorintheangleatE.Findthiserrorandfindthedimensionsofthepaperthatyouwouldneedtousetogetanaccurateregularpentagonandhenceanaccurateregulardodecahedron.HintJoinAO.WhatistheanglebetweenAOandtheredfoldline?NowfindangleREO.

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