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Mixed exercises
{{exercise_number}}. There are four different boxes in the image.
- Calculate the volume of each box.
- How many of the box \latex{ 1 } can be fitted into box \latex{ 3 }?
- How many of the box \latex{ 1 } can be fitted into box \latex{ 4 }?
\latex{ 4. }
\latex{ 1. }
\latex{ 2. }
\latex{ 3. }
The dimensions of each box:
1. \latex{ 250\times 175\times 100 } \latex{(mm)}
2. \latex{ 350\times 250\times 120 } \latex{(mm)}
3. \latex{ 400\times 250\times 150 } \latex{(mm)}
4. \latex{ 500\times 300\times 200 } \latex{(mm)}
1. \latex{ 250\times 175\times 100 } \latex{(mm)}
2. \latex{ 350\times 250\times 120 } \latex{(mm)}
3. \latex{ 400\times 250\times 150 } \latex{(mm)}
4. \latex{ 500\times 300\times 200 } \latex{(mm)}
{{exercise_number}}. Which of the following statements are true and which are false?
- Every square prism is a cuboid.
- There is a cuboid, which is not a square prism.
- There is no square prism that is a cube.
- There is no rectangular prism that is a cube.
- Not all cubes are cuboids.
{{exercise_number}}. Two edges of a cuboid are \latex{ 5 } and \latex{ 12 } units long. What is the smallest possible volume and surface area of this cuboid if the length of the third edge is also a whole number?
{{exercise_number}}. Three cubes with \latex{ 1 } \latex{ dm } long edges are put on top of each other. By how much does the surface area of the solid increase if another cube is placed on top? And if another one is placed there?
{{exercise_number}}. What is the surface area of the cubes with the following volumes?
a) \latex{ 1,000 } \latex{ cm^{3} }
b) \latex{ 27 } \latex{ m^{3} }
c) \latex{ 8 } \latex{ mm^{3} }
d) \latex{ 125 } \latex{ m^{3} }
e) \latex{ 64 } \latex{ mm^{3} }
{{exercise_number}}. What is the volume of the cubes with the following surface areas?
a) \latex{ 6 } \latex{ m^{2} }
b) \latex{ 150 } \latex{ cm^{2} }
c) \latex{ 54 } \latex{ m^{2} }
d) \latex{ 294 } \latex{ mm^{2} }
e) \latex{ 600 } \latex{ cm^{2} }
{{exercise_number}}. Compare the surface area and volume of the two solids shown in the image, knowing they consist of congruent cubes. (→)
b)
a)
{{exercise_number}}. Take cubes with a volume of \latex{ 1 } \latex{ cm^{3} } , \latex{ 8 } \latex{ cm^{3} } , and \latex{ 27 } \latex{ cm^{3} } , and glue them together so that the surface area of the resulting solid is the smallest possible. How large is this surface area?
{{exercise_number}}. A solid is built using \latex{ 11 } cubes. \latex{ 10 } cubes are already glued together, as shown in the image. Where should the last cube be placed so that the surface area of the resulting solid is the smallest possible? (→)
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{{exercise_number}}. The pedestal of a statue consists of \latex{ 14 } cubes with \latex{ 1 } \latex{ m } long edges (see image). The outside of the pedestal will be painted, except for the bottom. How many square metres do they have to paint? (→)
46044
{{exercise_number}}. How do the surface area and the volume of a cube with \latex{ 5 } \latex{ cm } long edges change if
a) a cube with \latex{ 1 } \latex{ cm } long edges is cut out at every vertex;
b) a cube with \latex{ 1 } \latex{ cm } long edges is cut out at the middle of every edge;
c) a cube with \latex{ 1 } \latex{ cm } long edges is placed at the centre of every face;
d) a cube with \latex{ 1 } \latex{ cm } long edges is cut out from the centre of each face?
b) a cube with \latex{ 1 } \latex{ cm } long edges is cut out at the middle of every edge;
c) a cube with \latex{ 1 } \latex{ cm } long edges is placed at the centre of every face;
d) a cube with \latex{ 1 } \latex{ cm } long edges is cut out from the centre of each face?
d)
a)
b)
c)
{{exercise_number}}. How can a cube with \latex{ 1 } \latex{ cm } long edges be cut out from a cube with \latex{ 5 } \latex{ cm } long edges in such a way that the surface area of the resulting solid is
a) \latex{ 158 } \latex{ cm^{2} } ; b) \latex{ 164 } \latex{ cm^{2}? }
Find various solutions.
{{exercise_number}}. A tetrahedron is cut at (→)
a) a vertex;
b) every vertex
of a cube with a plane that crosses the trisection point closest to the vertex. How many faces, edges and vertices do the resulting solids have?
b) every vertex
of a cube with a plane that crosses the trisection point closest to the vertex. How many faces, edges and vertices do the resulting solids have?
c) How many faces, edges and vertices do the resulting solids have if the planes cross the midpoint of every edge?
b)
a)
{{exercise_number}}. The edges of a cube are painted red and blue so that every face has at least one red edge. How many red edges will there be at least?
{{exercise_number}}.
- A cube with \latex{ 3 } \latex{ cm } long edges;
- a cuboid with \latex{ 3 } \latex{ cm }, \latex{ 4 } \latex{ cm } and \latex{ 5 } \latex{ cm } long edges
is dipped into paint and then cut into cubes with \latex{ 1 } \latex{ cm } long edges.
How many small cubes will there be with \latex{ 0 }, \latex{ 1 }, \latex{ 2 }, \latex{ 3 }, \latex{ 4 }, \latex{ 5 }, and \latex{ 6 } painted faces?
{{exercise_number}}. Three adjacent faces of a wooden cube with \latex{ 3 } \latex{ cm } long edges are painted red, while the rest are painted blue. Then, it is cut into cubes with \latex{ 1 } \latex{ cm } long edges using planes parallel to its faces. How many small cubes with different colour patterns have been created?
{{exercise_number}}. A cuboid consists of \latex{ 12 } unit cubes. One of the edges of this solid is twice as long as another edge, and the difference between two of its edges is \latex{ 1 } unit. How long are the edges of the cuboid?
{{exercise_number}}. Make a cuboid using \latex{ 12 } unit cubes, then come up with statements that allow your classmates to calculate the length of the edges of the solid.
{{exercise_number}}. In a set of building blocks, there is a circular hole in the middle of \latex{ 5 } faces of each block and a circular protrusion at the centre of one face. At least how many of these building blocks must be assembled so that no part is sticking out on the resulting solid? (→)
{{exercise_number}}. The image shows three different views of the same cube. (→)
face?
Which face is opposite this
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{{exercise_number}}. The sizes and colours of the three cubes are the same, but all six faces are painted with different colours. What colour is opposite white if Albert, Ben and Carl see their cubes this way when holding them in their hands?
C)
A)
B)
{{exercise_number}}. The image shows the dimensions of a fish tank. It is half-filled with water. When sand and pebbles are put into it, the water level rises \latex{ 6 } \latex{ cm }. (→)
- What is the total volume of the sand and pebbles in the fish tank?
- How high is the water level in the fish tank if an additional \latex{ 40 } \latex{ litres } of water is poured into it?
\latex{ 80 } \latex{ cm }
\latex{ 50 } \latex{ cm }
\latex{ 60 } \latex{ cm }
{{exercise_number}}. A two-digit number is created using a standard red and blue dice. The number on the blue dice will be in the tens place, and the red one in the ones place. How many two-digit numbers can be created? How many three-digit numbers can be made using a green, blue and red dice for the hundreds, tens and ones place, respectively?
Fun activity
The images below show solids that can form a cube if two are placed next to each other. Make similar solids using cardboard. Make as many as possible.
