SketchUp 3D Unit
1. Start with building prisms using the rectangle tool or the polygon tool and the Push-Pull tool
Mathematical Content
a) count vertices, edges and faces working towards Euler’s Theorem V-E+F = 2
b) calculate the TakeOut Angle at each vertex. This is equal to 360-sum of the adjacent face angles.
c) calculate surface area and volume.
d) Examine the symmetry operations and add all rotation axes. Color them in such a way that all axes with the same angle of rotation have the same color.
e) Create a net for your shape
f) Construct the dual which is called a dipyramid and repeat parts a) through e)
2. Construct a right pyramid which is a pyramid which has a regular polygon as the base and equilateral triangles as lateral faces.
a) count vertices, edges and faces working towards Euler’s Theorem V-E+F = 2
b) calculate the TakeOut Angle at each vertex. This is equal to 360-sum of the adjacent face angles.
c) calculate surface area and volume.
d) Examine the symmetry operations and add all rotation axes. Color them in such a way that all axes with the same angle of rotation have the same color.
e) Create a net for your shape
f) Construct the dual and repeat parts a) through e)
Construction Challenge.
Build a right pyramid with a pentagonal base 25” on a side with lateral faces that are 25 “ x ‘30” x 30”
3. The Platonic Solids
There are five of them: tetrahedron, cube, octahedron, dodecahedron, icosahedron
Here is a construction that will give you the first three.
a) First construct a cube.
b) Construct a tetrahedron inscribed in the cube.
c) Erase the edges of the cube
d) Consttruct the midtriangle of each face:i.e connect midpoints of three sides of the triangle.
e) Use the Eraser to erase the four vertices of the tetrahedron.
f) If you had kept a copy of each figure you can now reshape them using the measure tool and have three models all with edges of the same lengths.
For each model you can now do the following.
a) count vertices, edges and faces working towards Euler’s Theorem V-E+F = 2
b) calculate the TakeOut Angle at each vertex. This is equal to 360-sum of the adjacent face angles.
c) calculate surface area and volume.
d) Examine the symmetry operations and add all rotation axes. Color them in such a way that all axes with the same angle of rotation have the same color.
e) Create a net for your shape
f) Construct the dual and repeat parts a) through e)
Here is another way to constuct the dual which we will use later on.
1. Construct all the axes of symmetry for an octahedon. Those that go through the vertices are axes for rotations of 90 degrees. Those through the centers of the faces are axes for 120 degrees.
2. Construct a line (green line) perpendicular to edge AB and segment OM where O is the center of the octahedron and M is the midpoint of edge AB
3. Rotate the green line about one of the 4 fold rotation axes.
4. The four green lines outline a square. Construct that square using the line tool.
5. Erase the green lines and color the square,
6. Using the axes of symmetry construct the rest of the square using rotations.
7. Erase the Cuboctahedron and you have its dual the Cube. Note that all he axs of symmetry for the octahedron are also axes of symmetry for the cube. This shows an important property of duality!!