Kori on tyhjä
Spatial symmetry (supplementary material)
Example 1
Build symmetrical and asymmetrical solids using \latex{ 8 }, then only \latex{ 7 } identical small cubes. Make a drawing.
Solution
Symmetrical solids:
\latex{ 8 } cubes
\latex{ 7 } cubes
Asymmetrical solids:
\latex{ 8 } cubes
\latex{ 7 } cubes
Example 2
Build the following solids, shown from different views, using small cubes. Make drawings as well. Determine whether they are symmetrical or asymmetrical.
A)
front
view
view
top
view
view
side
view
view
B)
front
view
view
top
view
view
side
view
view
front view
side view
top view
Solution
A) symmetrical
B) asymmetrical
Example 3
Complete the solid shown in the image, so that it becomes symmetrical. Find several solutions.
What is the surface area and the volume of the resulting shape if the length of the edges of a small cube is \latex{ 2 \;cm ?}
Solution
By adding one small cube, you can create various symmetrical solids:
A)
B)
C)
All of these solids consist of \latex{ 6 } small cubes, so their total volume is:
\latex{ V=6\times 2\times 2\times 2 } \latex{ cm^{3}=48 } \latex{ cm^{3} }.
The faces of the solids consist of small squares. Count the number of squares.
The area of a small square is \latex{ 2\times 2=4 } \latex{ cm^{2} }. Based on this, you can calculate the surface area of the solid.
The area of a small square is \latex{ 2\times 2=4 } \latex{ cm^{2} }. Based on this, you can calculate the surface area of the solid.
- The number of squares:
\latex{ 2+2+2+2+4+4+3+3=22. }
Therefore, the surface area of the solid is \latex{ 22\times 4=88 } \latex{ cm^{2} }.
- The solid is the same as A), lying on a different face, so its surface area is also \latex{ 88 } \latex{ cm^{2} }.
- The number of squares of the faces
\latex{ 4+1+1+1+3+2+2+3+1+1+1+4=24.}
Therefore, the surface area of the solid is \latex{ 24\times 4=96 } \latex{ cm^{2} }.
Exercises
{{exercise_number}}. Build symmetrical and asymmetrical solids using \latex{ 9 }, then \latex{ 10 } small cubes. Make a drawing of them.
{{exercise_number}}. In the image, you can see the 'half' of symmetrical solids. Build the other half and make a drawing of the solids. Find several solutions.
a)
b)
c)
{{exercise_number}}. Stick together two identical rectangular cuboids in such a way to get
- a symmetrical solid;
- an asymmetrical solid.
{{exercise_number}}. The image shows two views of the same solid. Build a symmetrical solid using the fewest cubes possible, corresponding to the views. Draw the third view of the solid. What is the volume of the resulting solid if the edges of a cube are \latex{ 3 \;cm } long?
a)
b)
{{exercise_number}}. These apple 'halves' are the mirror images of each other. Can you put them together to get a complete apple?
{{exercise_number}}. Remove the fewest cubes from the following solids to create symmetrical shapes. Make a drawing of the resulting solids. What is their surface area if the edges of a cube are \latex{ 4 \,cm } long?
a)
b)
c)
{{exercise_number}}. The first image shows the net of a matchbox. Which is the net of the drawer?
matchbox
drawer
{{exercise_number}}. Glue together two dice along their faces. How many different sums can be on the resulting dice?
{{exercise_number}}. Gale has \latex{ 8 } identical blue cubes. At least how many faces should Zoe paint red, so that Gale cannot make a large, entirely blue cube?
{{exercise_number}}. The image shows a matchbox. How far apart are the two edges marked with red from each other when the box is flattened?
\latex{ 1.4 } \latex{ cm }
\latex{ 5.5 } \latex{ cm }
\latex{ 3.6 } \latex{ cm }
{{exercise_number}}. The cube shown below is wrapped in coloured paper. Which net corresponds to the wrapping?
Quiz
Which direction is the bus moving in? What do you think?
