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The volume of cuboids
Example
How many boxes can be loaded into the lorry if the cargo space measures \latex{3} \latex{metres} by \latex{2} \latex{metres} by \latex{8} \latex{metres?}
Solution
\latex{ 3 } boxes can be put next to each other, and on top of each, it is possible to place one more box, which means a total of \latex{ 3\times2 = 6 } boxes. Since the cargo space is \latex{ 8 } \latex{ m } long, \latex{ 8 } similar rows can be placed next to each other. So \latex{ 8\times6 = 48 } boxes can be loaded into the lorry.
\latex{3 \times 2 = 6} boxes
\latex{(3 \times 2) \times 8 = 48} boxes
The boxes can also be loaded in the following ways:
\latex{2\times8=16} boxes
\latex{(2\times8)\times3=48} boxes
\latex{8\times3=24} boxes
\latex{(8\times3)\times2=48} boxes
You can notice that the result will be the same regardless of the order in which the boxes are loaded.
\latex{(3 \times 2) \times 8 = (2 \times 8) \times 3 = (8 \times 3) \times 2 = 48}
If the lengths of the edges of a cuboid are whole numbers, its volume can be determined by calculating the number of unit cubes that can be fitted inside it. The symbol of volume is \latex{V}.
Calculating the volume of a cuboid
Observe a cuboid with \latex{ 3 } \latex{ cm }, \latex{ 4 } \latex{ cm }, and \latex{ 5 } \latex{ cm } long edges. Its volume is determined by filling it with cubes with \latex{ 1 } \latex{ cm } long edges.
- Volume unit: the volume of a cube with \latex{ 1 } \latex{ cm } long edges (\latex{ 1 } \latex{ cm^{3} }).
- Along the \latex{ 3 } \latex{ cm } long edge, \latex{ 3 } cubes can be fitted.
- \latex{ 3\times5 } cubes can be fitted on the same level.
- The cuboid can contain \latex{ 4 } levels of cubes.
\latex{ 4 } \latex{ cm }
\latex{ 3 } \latex{ cm }
\latex{ 5 } \latex{ cm }
658051
The volume of the cuboid with \latex{ 3 } \latex{ cm }, \latex{ 5 } \latex{ cm } and \latex{ 4 } c\latex{ m } long edges is:
\latex{V = 3 \times 5 \times 4\;cm ^{3} = 60\;cm^{3}.}
The volume of a cuboid is the product of the length of the edges meeting at a vertex. The volume of a cuboid with a, b and c edges:
\latex{V= a\times b \times c}.
(This formula is also true if the lengths of the edges are not whole numbers.)
\latex{ c }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ c }
\latex{ a }
\latex{ b }
\latex{ c }
\latex{V= a\times b\times c}
\latex{V= a\times c\times b}
\latex{V= b\times c\times a}
If you calculate the volume of a cuboid by turning it on three different faces, you get the same result in each case.
The volume of a cuboid can be calculated by multiplying the area of a face (base) by the length of an edge perpendicular to it (height).
The volume of cubes
Build a cube with \latex{ 5 } \latex{ cm } long edges using cubes with \latex{ 1 } \latex{ cm } long edges.
- There are \latex{ 5\times5 } small cubes on one level.
- The large cube contains \latex{ 5 } levels of small cubes.
\latex{ 5 } \latex{ cm }
\latex{ 5 } \latex{ cm }
\latex{ 5 } \latex{ cm }
The volume of the cube with \latex{ 5 } \latex{ cm } long edges is:
\latex{V = 5 \times 5 \times 5\;cm^{3} = 125\;cm^{3}}.
The volume of a cube with \latex{a} edges is:
\latex{V = a \times a \times a}.
Exercises
{{exercise_number}}. Calculate the volume of the following cube-shaped solids.
a) sugar cube with \latex{ 12 } \latex{ mm } long edges
b) magic cube with \latex{ 6 } \latex{ cm} long edges
c) lift with \latex{ 25 } \latex{ dm } long edges
d) room with \latex{ 300 } \latex{ cm } long edges
{{exercise_number}}. How large will the volume of a cube with \latex{ 3 } \latex{ cm } long edges be if
a) the length of the edges is doubled;
b) the length of the edges is tripled?
b) the length of the edges is tripled?
How many times larger is the volume of the new cube than that of the original cube? (⟶)
a)
b)
{{exercise_number}}. Calculate the volume of the cube if its edges are
a) \latex{ 96 } \latex{ mm };
b) \latex{ 60 } \latex{ cm };
c) \latex{ 108 } \latex{ cm };
d) \latex{ 360 } \latex{ mm }.
{{exercise_number}}. A cube was built using six squares. What is the volume of the cube if the area of a square is
a) \latex{ 4 } \latex{ cm^{2} };
b) \latex{ 121 } \latex{ mm^{2} };
c) \latex{ 160, 000 } \latex{ mm^{2} };
d) \latex{ 36 } \latex{ cm^{2} ?}
{{exercise_number}}. Can the cubes with the following volumes fit in your pocket?
a) \latex{ 8,000 } \latex{ mm^{3} }
b) \latex{ 64 } \latex{ cm^{3} }
c) \latex{ 343 } \latex{ cm^{3} }
{{exercise_number}}. Can a cube with a volume of \latex{ 27 } \latex{ m^{3} } fit through the door of your classroom?
{{exercise_number}}. The solids shown in the image were built using congruent cubes with \latex{ 1 } \latex{ cm } long edges. (⟶)
a) What is their volume?
b) At least how many small cubes are needed to turn them into a cube?
b) At least how many small cubes are needed to turn them into a cube?
\latex{ A })
\latex{ B })
{{exercise_number}}. A cube with \latex{ 2 } \latex{ cm } long edges was cut out of a cube with \latex{ 3 } \latex{ cm } long edges. (⟶)
a) What is the volume of the resulting solid?
b) How did the surface area of the solid change?
b) How did the surface area of the solid change?
\latex{ 8. }
{{exercise_number}}. How many different cuboids can be built using cubes with \latex{ 1 } \latex{ cm } long edges? What is their volume? Which one has the smallest surface area?
{{exercise_number}}. A cube with \latex{ 2 } \latex{ cm } long edges is cut out of a cube with \latex{ 5 } \latex{ cm } long edges, as shown in the image. The part that is cut out is then placed on top of the solid. (⟶)
a) What is the volume of the resulting solid?
b) How did the surface area of the solid change?
b) How did the surface area of the solid change?
\latex{ 10. }
{{exercise_number}}. Can \latex{ 8, } \latex{ 9 } and \latex{ 10 } cubes with \latex{ 1 } \latex{ cm } long edges be cut out of a cube with \latex{ 5 } \latex{ cm } long edges without modifying its volume?
{{exercise_number}}. Calculate the volume of the cuboid if its edges are
a) \latex{ 2 } \latex{ mm }; \latex{ 7 } \latex{ mm } and \latex{ 8} \latex{ mm };
b) \latex{ 15 } \latex{ cm }; \latex{ 15 } \latex{ cm } and \latex{ 56 } \latex{ cm };
c) \latex{ 32 } \latex{ m }; \latex{ 5 } \latex{ m } and \latex{ 8 } \latex{ m };
d) \latex{ 102 } \latex{ m }; \latex{ 7 } \latex{ m } and \latex{ 7 } \latex{ m };
e) \latex{ 200 } \latex{ mm }; \latex{ 20 } \latex{ cm } and \latex{ 2 } \latex{ m };
f) \latex{ 3 } \latex{ m }; \latex{ 40 } \latex{ cm } and \latex{ 90 } \latex{ cm };
g) \latex{ 10 } \latex{ mm }; \latex{ 1 } \latex{ cm } and \latex{\frac{1}{100}} \latex{ m };
h) \latex{ 4 } \latex{ m }; \latex{ 4 } \latex{ mm } and \latex{ 4 } \latex{ cm }.
{{exercise_number}}. Calculate the volume of the cuboid using the following information (the first value is the area of a face; the second value is the length of the edge perpendicular to it).
a) \latex{ 35 } \latex{ cm^{2} }; \latex{ 18 } \latex{ cm };
b) \latex{ 197 } \latex{ cm^{2} }; \latex{ 23 } \latex{ cm };
c) \latex{ 48,000 } \latex{ cm^{2} }; \latex{ 3 } \latex{ cm };
d) \latex{ 719 } \latex{ mm^{2} }; \latex{ 5 } \latex{ cm };
e) \latex{ 3,700 } \latex{ mm^{2} }; \latex{ 42 } \latex{ cm };
f) \latex{ 150 } \latex{ m^{2} }; \latex{ 38 } \latex{ cm }.
{{exercise_number}}. You can see two views of a cuboid in the image. Calculate the length of its edges and its volume.
\latex{ 1 }
\latex{ 4 }
\latex{ 6 }
\latex{ 5 }
\latex{ 2 }
a)
b)
\latex{ 5 }
{{exercise_number}}. Which of the following cuboids has the largest volume? First, use only estimation, and then perform actual calculations.
a)
\latex{ 7 }
\latex{ 6 }
\latex{ 8 }
\latex{ 2 }
\latex{ 3 }
\latex{ 20 }
\latex{ 2 }
\latex{ 2 }
\latex{ 10 }
b)
\latex{ 1 }
\latex{ 1 }
\latex{ 36 }
\latex{ 2 }
\latex{ 3 }
\latex{ 6 }
\latex{ 1 }
\latex{ 4 }
\latex{ 9 }
{{exercise_number}}. Find cuboids with a volume of \latex{ 64 } \latex{ cm^{3} } whose edges in \latex{ centimetres } are whole numbers. Make the net of one of them and fold it into a solid.
{{exercise_number}}. Calculate the length of the edge perpendicular to a face of the cuboid using its volume and the area of the face.
a) \latex{ 11, 664 } \latex{ cm^{3} }; \latex{ 486 } \latex{ cm^{2} }
b) \latex{ 3,588 } \latex{ cm^{3} }; \latex{ 6,900 } \latex{ mm^{2} }
c) \latex{ 1, 720, 000 } \latex{ mm^{3} }; \latex{ 215 } \latex{ cm^{2} }
{{exercise_number}}. Calculate the area of the face perpendicular to an edge using the volume of the cuboid and the length of the edge.
a) \latex{ 23, 688 } \latex{ cm^{3} }; \latex{ 56 } \latex{ cm }
b) \latex{ 2,006 } \latex{ cm^{3} }; \latex{ 340 } \latex{ mm }
c) \latex{ 1, 512, 000 } \latex{ cm^{3} }; \latex{ 4,200 } \latex{ mm }
{{exercise_number}}. \latex{ 15 } \latex{ cm } of snow fell on the \latex{ 20 } \latex{ m } long pavement outside your house. How many \latex{ cubic } \latex{ metres } of snow do you have to shovel if the pavement is \latex{ 1 } \latex{ m } wide?
{{exercise_number}}. Measure the dimensions of a cuboid-shaped juice box with a volume of \latex{ 1 } \latex{ litre }. How many \latex{ cubic } \latex{ millimetres } will remain empty after pouring \latex{ 1 } \latex{ litre } of juice in it? (Omit the thickness of the box's side.)
Quiz
Is there a solid with only square faces that is not a cube?
