Сагс хоосон байна
The cuboid
Based on their geometric properties, the solids in the drawing are cuboids.
vertex
face
edge
Properties of cuboids
– Cuboids have \latex{ 6 } faces, \latex{ 12 } edges and \latex{ 8 } vertices.
– All faces of cuboids are rectangles.
– Every edge belongs to two faces; these faces are adjacent.
– Adjacent faces are perpendicular to each other.
– \latex{ 3 } faces and \latex{ 3 } edges meet at every vertex.
– The edge pairs meeting at a vertex are perpendicular to each other.
– The opposite faces of cuboids are parallel and congruent.
– All faces of cuboids are rectangles.
– Every edge belongs to two faces; these faces are adjacent.
– Adjacent faces are perpendicular to each other.
– \latex{ 3 } faces and \latex{ 3 } edges meet at every vertex.
– The edge pairs meeting at a vertex are perpendicular to each other.
– The opposite faces of cuboids are parallel and congruent.
Observe the line segments connecting the vertices.
An edge connects adjacent vertices. The line segment connecting two non-adjacent vertices found on the same face is called a face diagonal. The line segment connecting two vertices that are not on the same face is called a space diagonal.
space diagonal
face
diagonal
diagonal
Classification of cuboids
Square prisms are cuboids that have two square faces.
Cubes are cuboids that have only square faces.
Cubes are cuboids that have only square faces.
Cubes
Cuboids
Square prisms
Exercises
{{exercise_number}}. The sum of the dots on opposing faces of a standard die is always \latex{ 7 }. Observe a standard die.
a) What is the sum of the dots on the adjacent faces of the \latex{ 1 } face?
b) What is the sum of the dots on the adjacent faces of the \latex{ 2 } face?
b) What is the sum of the dots on the adjacent faces of the \latex{ 2 } face?
{{exercise_number}}. How many dots are on the face parallel to the
a) \latex{ 3 };
b) \latex{ 6 };
c) \latex{ 5 };
d) \latex{ 4 } face?
{{exercise_number}}. The \latex{ 1 } and \latex{ 2 } faces of a die are perpendicular to each other.
a) Find all the faces that are perpendicular to the \latex{ 6 } face.
b) Find all the faces that are perpendicular to the \latex{ 4 } face.
b) Find all the faces that are perpendicular to the \latex{ 4 } face.
{{exercise_number}}. Choose three faces on a die and find faces that are perpendicular to them.
{{exercise_number}}. How many faces of the cuboid shown in the image are
a) parallel;
b) perpendicular;
c) congruent to the face coloured with red? (→)
b) perpendicular;
c) congruent to the face coloured with red? (→)
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ H }
\latex{ E }
\latex{ F}
\latex{ G}
{{exercise_number}}. List edges that are equal to the edge \latex{ AB }. What is their position compared to \latex{ AB? } (→)
\latex{ F}
\latex{ A}
\latex{ D}
\latex{ C}
\latex{ B}
\latex{ E}
\latex{ H}
\latex{ G}
{{exercise_number}}. The blue lines in the image show the edges of a cube. Find lines that are
a) parallel;
b) skew to line \latex{\large a}. (→)
b) skew to line \latex{\large a}. (→)
\latex{ j }
\latex{ e }
\latex{ g }
\latex{ c }
\latex{ a }
\latex{ h }
\latex{ k }
\latex{ l }
\latex{ h }
\latex{ i }
\latex{ d }
\latex{ f }
{{exercise_number}}. Choose four edges and find edges parallel to them.
{{exercise_number}}. Which of the following statements are true and which are false?
a) The number of edges of cuboids is twice the number of faces.
b) Cuboids have the same number of edges and vertices.
c) The sum of the number of vertices and faces of cuboids is \latex{ 2 } greater than the number of edges.
d) Only one face is not adjacent to any chosen face of the cuboid.
b) Cuboids have the same number of edges and vertices.
c) The sum of the number of vertices and faces of cuboids is \latex{ 2 } greater than the number of edges.
d) Only one face is not adjacent to any chosen face of the cuboid.
{{exercise_number}}. A square prism is cut in half through the vertices \latex{ A }, \latex{ B }, \latex{ C }, and \latex{ D }. How many faces, edges and vertices does the resulting solid have? (→)
\latex{ D }
\latex{ A }
\latex{ B }
\latex{ C }
{{exercise_number}}. Which of the following statements are true and which are false?
a) A cuboid has a face which is not a rectangle.
b) A cuboid can have edges with three different lengths.
c) There is a cuboid which has exactly three square faces.
d) There is a cuboid that has four edges with different lengths.
e) There is a cuboid which has exactly four square faces.
b) A cuboid can have edges with three different lengths.
c) There is a cuboid which has exactly three square faces.
d) There is a cuboid that has four edges with different lengths.
e) There is a cuboid which has exactly four square faces.
{{exercise_number}}. Solids are made using congruent cubes by placing congruent square faces on top of each other.
a) How many solids can be made using \latex{ 3 } congruent cubes?
b) How many can be made using \latex{ 4 } congruent cubes?
b) How many can be made using \latex{ 4 } congruent cubes?
{{exercise_number}}. A cube with \latex{ 4 }-unit-long edges was built using \latex{ 64 } unit cubes. A \latex{ 2 }-unit-wide strip was painted on it. How many unit cubes were not painted? (→)
{{exercise_number}}. Cuboids were built using unit cubes.
How many unit cubes were used?
Build these solids using sugar cubes. (→)
How many unit cubes were used?
Build these solids using sugar cubes. (→)
c)
a)
b)
Unit cube:
A cube with one unit long edges is used when the unit of measurement is irrelevant in the given exercise.
{{exercise_number}}. The letters mark the vertices in the figure. (→)
Example: edge \latex{ CB }, face diagonal \latex{ CA }, space diagonal \latex{ CE }.
List other edges, face diagonals and space diagonals.
Example: edge \latex{ CB }, face diagonal \latex{ CA }, space diagonal \latex{ CE }.
List other edges, face diagonals and space diagonals.
\latex{ G }
\latex{ A }
\latex{ E }
\latex{ H }
\latex{ D }
\latex{ F }
\latex{ B }
\latex{ C }
face diagonal
edge
space
diagonal
diagonal
{{exercise_number}}. How many space diagonals does a cuboid have?
{{exercise_number}}. How many face diagonals does a cuboid have? How many can have different lengths?
{{exercise_number}}. What is the sum of the number of edges, face diagonals and space diagonals of a cuboid?
{{exercise_number}}. Is there a cuboid that has only \latex{ 2 } \latex{ cm } and \latex{ 4 } \latex{ cm } long edges?
{{exercise_number}}. Find parallel face diagonals on the
a) cube;
b) cuboid. (→)
b) cuboid. (→)
\latex{ B }
a)
b)
\latex{ I }
\latex{ M }
\latex{ N }
\latex{ J }
\latex{ K }
\latex{ L }
\latex{ P }
\latex{ O }
\latex{ G }
\latex{ C }
\latex{ E }
\latex{ H }
\latex{ F }
\latex{ D }
\latex{ A }
{{exercise_number}}. All the vertices of a cuboid were connected with all the other vertices. How many line segments did you get?
{{exercise_number}}. Go through all the vertices of the cube shown in the previous exercise. You can move only along the edges and cannot cross the same vertex twice. (Write down your path using the letters of the vertices.)
{{exercise_number}}. A ribbon was tied around a cuboid-shaped package in two ways. How long does the ribbon have to be for the two packages if the edges of the cuboid are \latex{ 32 } \latex{ cm }, \latex{ 24 } \latex{ cm } and \latex{ 15 } \latex{ cm } long, and \latex{ 20 } \latex{ cm } of ribbon is used for the bow? (→)
b)
\latex{ 15 }
\latex{ 15 }
\latex{ 32 }
\latex{ 32 }
\latex{ 24 }
\latex{ 24 }
a)
{{exercise_number}}. At least how many colours do you need to colour a cube so that each face is painted with one colour and the adjacent faces have different colours?
{{exercise_number}}. \latex{ 1 } is written on each vertex of a cube. Then, the sum of the numbers at each endpoint is written on each edge, and the sum of the numbers of the surrounding edges is written on each face. What is the sum of the numbers on the faces?
{{exercise_number}}. \latex{ 2 } is written on two adjacent vertices of a cube, while \latex{ 1 } is written on all the other vertices. Then, the sum of the numbers at each endpoint is written on each edge, and the sum of the numbers of the surrounding edges is written on each face. What is the sum of the numbers on the faces?
Quiz
Using a ruler, how can you measure the distance of the two vertices found the furthest from each other, that is, the space diagonal?
