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The surface area of cuboids
Example 1
The edges of a paper cuboid meeting at a vertex are \latex{ 3 } \latex{ cm }, \latex{ 4 } \latex{ cm }, and \latex{ 5 } \latex{ cm } long. Unfold the solid along the edges. What is the area of the net?
Solution
\latex{ 5 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 3 } \latex{ cm }
\latex{3 \times 4 \text{ cm}^2}
\latex{4 \times 5 \text{ cm}^2}
\latex{3 \times 5 \text{ cm}^2}
\latex{3 \times 5 \text{ cm}^2}
\latex{4 \times 5 \text{ cm}^2}
\latex{3 \times 4 \text{ cm}^2}
Add the areas of the faces.
The opposite faces of the cuboid are congruent, thus:
\latex{2 \times (3\times4 \text{ cm}^2 + 4 \times 5 \text{ cm}^2 + 3 \times 5 \text{ cm}^2) = 94 \text{ cm}^2}
Example 2
How large is the surface a toy maker must sand if he makes \latex{ 100 } wooden cubes with
\latex{ 2 } \latex{ cm } long edges?
Solution
The surface area of a wooden cube is the sum of the areas of the faces, that is, the area of the net.
\latex{2 \times 2 \text{ cm}^2}
\latex{2 \times 2 \text{ cm}^2}
\latex{2 \times 2 \text{ cm}^2}
\latex{2 \times 2 \text{ cm}^2}
\latex{2 \times 2 \text{ cm}^2}
\latex{2 \times 2 \text{ cm}^2}
\latex{ 2 } \latex{ cm }
\latex{ 2 } \latex{ cm }
\latex{ 2 } \latex{ cm }
The cube is bordered by \latex{ 6 } congruent faces, thus
\latex{ 100 } pcs
\latex{6 \times 2 \times 2 \text{ cm}^2 = 6 \times 4 \text{ cm}^2 = 24 \text{ cm}^2}
The surface of a cube is \latex{ 24 } \latex{ cm^{2} } , so when sanding \latex{ 100 } wooden cubes, a surface area of \latex{100 \times 24\text{ cm}^2=2,400\text{ cm}^2} must be sanded.
The surface area of a solid bordered by plane figures is the sum of the areas of the faces. Its symbol is: \latex{A}.
The surface area of cuboids and cubes
The edges of the cuboid below are \latex{\large a}, \latex{\large b } and \latex{\large c } long, while the edges of the cube are a long.
\latex{ a }
\latex{ b }
\latex{ c }
\latex{ b \times c }
\latex{ a \times c }
\latex{ a \times c }
\latex{ a \times b }
\latex{ a \times b }
\latex{ b \times c }
The surface area of the cuboid with edges that are \latex{ a, b } and \latex{ c } long:
\latex{ A=2\times (a\times b+a\times c+b\times c) }.
\latex{ A=2\times (a\times b+a\times c+b\times c) }.
\latex{ a }
\latex{ a }
\latex{ a }
\latex{a\times a}
\latex{ a\times a }
\latex{ a \times a }
\latex{ a \times a }
\latex{ a \times a }
\latex{ a \times a }
The surface area of the cube with edges that are a long:
\latex{ A=6\times a\times a }.
\latex{ A=6\times a\times a }.
The surface area of a cuboid is twice the sum of the areas of the three faces meeting at a vertex.
The surface area of a cube is six times the area of a face.
The surface area of a cube is six times the area of a face.
\latex{ 2\times (a\times b+a\times c+b\times c) }
The units of surface area are the same as those of area.
\latex{1\;{ mm}^2 \underset{\times 100}{\lt } 1\;{ cm}^2 \underset{\times 100}{\lt } 1\;{ dm}^2 \underset{\times 100}{\lt } 1\;{ m}^2 \underset{\times 1,000,000}{\lt } 1\;{ km}^2}
Example 3
Vilma covered her cuboid-shaped box by glueing a coloured sheet of paper on each side. A sheet of paper covered the side completely.
\latex{ 6 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 4 } \latex{ cm }
a) What should the size of the colourful papers be? Draw them and colour the same ones with the same colour.
b) What is the sum of the surfaces of the colourful papers?
Solution
Write the length of each edge on the drawing, knowing that every face of the cuboid is a rectangle and its opposite faces are equal.
\latex{ 6 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 4 }
\latex{ 6 }
\latex{ 6 }
\latex{ 6 }
\latex{ 4 }
\latex{ 4 }
\latex{ 4 }
\latex{ 4 }
\latex{ 4 }
a) The faces of the cuboid:
\latex{ 4 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 6 } \latex{ cm }
\latex{ 6 } \latex{ cm }
\latex{ 6 } \latex{ cm }
\latex{ 6 } \latex{ cm }
\latex{ 4 } \latex{ cm }
b) The sum of the areas of the pieces of paper is the surface area of the cuboid:
\latex{\text{A} = 2 \times (4 \times 4) + 4 \times (4 \times 6)= 128 (\text{cm}^2)}
Exercises
{{exercise_number}}. Calculate the surface area 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{ 250 } \latex{ cm } long edges
d) room with \latex{ 300 } \latex{ cm } long edges
{{exercise_number}}. Calculate the surface area of the cube with
a) \latex{ 4 } \latex{ cm };
b) \latex{ 10 } \latex{ m };
c) \latex{ 8 } \latex{ m };
d) \latex{ 1,700 } \latex{ mm };
e) \latex{ 89 } \latex{ m };
f) \latex{ 400 } \latex{ mm };
g) \latex{ 2 } \latex{ cm } \latex{ 3 } \latex{ mm };
h) \latex{ 3 } \latex{ m } \latex{ 5 } \latex{ cm } long edges.
{{exercise_number}}. What is the surface area of the cube if the area of its faces is
a) \latex{ 9 } \latex{ cm^{2} };
b) \latex{ 49 } \latex{ m^{2} } ;
c) \latex{ 81 } \latex{ mm^{2} ?}
{{exercise_number}}. What is the surface area of the cube if
a) the sum of the area of opposite faces is \latex{ 96 } \latex{ cm^{2} };
b) the sum of the area of four faces is \latex{ 76 } \latex{ mm^{2} };
c) the sum of the areas of six faces is \latex{ 234 } \latex{ m^{2}? }
b) the sum of the area of four faces is \latex{ 76 } \latex{ mm^{2} };
c) the sum of the areas of six faces is \latex{ 234 } \latex{ m^{2}? }
{{exercise_number}}. How large will the surface area of a cube with \latex{ 3 } \latex{ cm } 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 surface area of the new cube than that of the original cube? (→)
a)
b)
{{exercise_number}}. What is the surface area of the cube if the total length of all its edges is
a) \latex{ 96 } \latex{ mm };
b) \latex{ 60 } \latex{ cm };
c) \latex{ 108 } \latex{ cm };
d) \latex{ 360 } \latex{ mm ?}
{{exercise_number}}. What is the area of a face if the total surface area of the cube is
a) \latex{ 48 } \latex{ cm^{2} };
b) \latex{ 66 } \latex{ mm^{2} };
c) \latex{ 96 } \latex{ m^{2} };
d) \latex{ 75 } \latex{ cm^{2} ?}
{{exercise_number}}. How long are the edges of a cube if its surface area is
a) \latex{ 24 } \latex{ m^{2} };
b) \latex{ 54 } \latex{ cm^{2} };
c) \latex{ 600 } \latex{ m^{2} };
d) \latex{ 150 } \latex{ mm^{2} ?}
{{exercise_number}}. Can a cube with a surface area of \latex{ 15,000 } \latex{ cm^{2} } fit through the door of the classroom?
{{exercise_number}}. Can the cubes with the following surface areas fit in your pocket?
a) \latex{ 24 } \latex{ cm^{2} }
b) \latex{ 60,000 } \latex{ mm^{2} }
c) \latex{ 150 } \latex{ cm^{2} }
{{exercise_number}}. A cube with \latex{ 3 } \latex{ cm } long edges was cut out of a cube with \latex{ 6 } \latex{ cm } long edges, as shown in the image. What is the surface area of the resulting solid? (→)
{{exercise_number}}. A cube with \latex{ 1 } \latex{ cm } long edges was glued on top of a cube with
\latex{ 3 } \latex{ cm } long edges, as shown in the image. What is the surface area of the resulting solid? (→)
{{exercise_number}}. Calculate the surface area of the cuboid if its edges are
a) \latex{ 2 } \latex{ mm }; \latex{ 7 } \latex{ mm }; \latex{ 8 } \latex{ mm };
b) \latex{ 15 } \latex{ cm }; \latex{ 39 } \latex{ cm }; \latex{ 56 } \latex{ cm };
c) \latex{ 32 } \latex{ m }; \latex{ 5 } \latex{ m }; \latex{ 8 } \latex{ m };
d) \latex{ 102 } \latex{ cm }; \latex{ 7 } \latex{ m }; \latex{ 70 } \latex{ mm };
e) \latex{ 18 } \latex{ cm }; \latex{ 120 } \latex{ mm }; \latex{ 400 } \latex{ mm };
f) \latex{ 30 } \latex{ m }; \latex{ 400 } \latex{ cm }; \latex{ 5,000 } \latex{ mm };
g) \latex{ 125 } \latex{ cm }; \latex{ 3 } \latex{ m }; \latex{ 300 } \latex{ mm };
h) \latex{ 5,600 } \latex{ mm }; \latex{ 480 } \latex{ cm }; \latex{ 2 } \latex{ m }.
{{exercise_number}}. Determine the surface area of the cuboid if the areas of the faces meeting at a vertex are
- \latex{ 12 } \latex{ cm^{2} }; \latex{ 18 } \latex{ cm^{2} }; \latex{ 24 } \latex{ cm^{2} };
- \latex{ 56 } \latex{ cm^{2} }; \latex{ 7 } \latex{ cm^{2} }; \latex{ 8 } \latex{ cm^{2} };
- \latex{ 1 } \latex{ m^{2} }; \latex{ 2 } \latex{ m^{2} }; \latex{ 32 } \latex{ m^{2} };
- \latex{ 1,600} \latex{ cm^{2} }; \latex{ 4,000 } \latex{ cm^{2} }; \latex{\frac{1}{10} } \latex{ m^{2} }.
{{exercise_number}}. The image shows two views of a cuboid. Calculate the length of the edges of the cuboid and its surface area if the side of a square is \latex{ 2 } \latex{ cm }.
a)
b)
front view
side view
front view
side view
{{exercise_number}}. Two of the edges of a cuboid meeting at a vertex are equal and one is \latex{ 2 } \latex{ cm } longer than the other two. The sum of these edges is \latex{ 14 } \latex{ cm }. What is the surface area of the cuboid?
{{exercise_number}}. The edges of a \latex{ 25 } \latex{ m^{2} } room measured in \latex{ metres } are whole numbers, and its height is \latex{ 3 } \latex{ m }. It has two doors that are \latex{ 1 } \latex{ m } wide and \latex{ 2 } \latex{ m } high. It also has a \latex{ 3 } \latex{ m } wide and \latex{ 2 } \latex{ m } high window. The owners want to paint the room. How many \latex{ kg } of paint should they buy if \latex{\frac {1}{5}} \latex{ kg } is needed to paint one \latex{ square } \latex{ metre? }
{{exercise_number}}. A \latex{ 4 } \latex{ cm \times 5 } \latex{ cm \times 2 } \latex{ cm } cuboid was cut from the cuboid with \latex{ 4 } \latex{ cm }, \latex{ 5 } \latex{ cm } and \latex{ 10 } \latex{ cm } long edges, as shown in the image. By how many \latex{ square } \latex{ centimetres } did the surface area of the cuboid decrease? (→)
\latex{ 2 } \latex{ cm }
\latex{ 4 } \latex{ cm }
\latex{ 5 } \latex{ cm }
\latex{ 10 } \latex{ cm }
{{exercise_number}}. The edges of a red cuboid are \latex{ 2 } \latex{ cm }, \latex{ 4 } \latex{ cm } and \latex{ 7 } \latex{ cm }, while the edges of a blue cuboid are
\latex{ 2 } \latex{ cm }, \latex{ 4 } \latex{ cm } and \latex{ 10 } \latex{ cm }. The two cuboids are glued together so their congruent faces fit perfectly.
a) What is the surface area of the resulting solid?
b) How can the two solids be glued together so that the surface area of the resulting solid is the same as in the previous exercise?
b) How can the two solids be glued together so that the surface area of the resulting solid is the same as in the previous exercise?
{{exercise_number}}. How does the surface area of the cuboid in the image change if a cube with \latex{ 1 } \latex{ cm } edges is cut out at every vertex of the upper face? (→)
\latex{ 6 } \latex{ cm }
\latex{ 10 } \latex{ cm }
\latex{ 8 } \latex{ cm }
{{exercise_number}}. Three matchboxes were glued together to form a cuboid. What is the surface area of the resulting solid if the sides of a matchbox are \latex{ 52 } \latex{ mm }, \latex{ 35 } \latex{ mm}, and \latex{ 15 } \latex{ mm } long?
Quiz
Three standard dice were put next to each other, as shown in the image. What is the sum of all the visible dots?
