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Grade 10 Mathematics Teachers’ Book

INNODEMS

Subsection 2.7.1 (*) Surface Area of Cubes and Cuboids

Learner Experience 2.7.2.

Work in groups: Form groups of \(2\) or \(3\) students.
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Materials Needed:
  1. Solids cube, or cuboid wood waste blocks
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  2. Grid/graph paper or plain paper
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  3. Rulers
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  4. Pre-made nets of cubes or cuboids (optional)
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Instructions:
  1. Choose one solid object (cube or cuboid).
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  2. Create a net for the chosen object (either by unfolding a model or using drawn net templates).
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  3. Trace the faces onto grid paper or measure them using a ruler or string.
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  4. Calculate the area of each face.
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  5. Add up all face areas.
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What will happen if the surface area doubles in size?
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Why do bigger cubes have more surface area?
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Key Takeaway 2.7.2.

The surface area of a cube or cuboid is the total area of all its faces. It can be calculated by finding the area of each face and then adding them together.
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A cube has \(6\) identical square faces. If it has a side length of \(s\text{,}\) then the area of each face is \(s^2\text{,}\) so we have:
\begin{equation*} \text{Surface Area of a Cube} = 6s^2 \end{equation*}
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A cuboid has 6 rectangular faces. If \(l\) = length, \(w\) = width, \(h\) = height, then:
\begin{equation*} \text{Surface Area of a Cuboid} = 2(lw + lh + wh) \end{equation*}
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A \(\textbf{cuboid}\) is also called a rectangular prism.
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Example 2.7.3.

Work out the surface area of a cube whose side is \(12 \ \text{cm}\text{.}\)
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Solution.
Using the formula, with \(s = 12 \, \text{cm}\text{,}\) we have:
\begin{align*} \text{Surface Area} \amp = 6s^2 \\ \amp = 6(12^2) \\ \amp = 6(144) \\ \amp = 864 \text{ cm}^2 \end{align*}
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Example 2.7.4.

Find the surface area of the following cuboid:
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Solution.
Sketch and label the net of the cuboid
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Find the areas of the different shapes in the net
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\begin{align*} \text{large rectangle} = \amp \text{ perimeter of small rectangle} \times \text{length} \\ = \amp (3+8+3+8)\text{cm} \times 14\text{cm} \\ = \amp 22\text{cm} \times 14\text{cm} \\ = \amp 308\text{cm}^2 \end{align*}
\begin{align*} 2 \, \text{small rectangle} \amp = 2(8 \text{cm} \times 3\text{cm} ) \\ = \amp 2(18) \,\text{cm} \\ = \amp 36\text{cm}^2 \end{align*}
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Find the sum of the areas of the faces
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\begin{align*} \text{large rectangle} + \text{small rectangle} = \amp (308 + 36)\text{cm}^2 \\ = \amp 344 \, \text{cm}^2 \end{align*}
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The surface area of the cuboid is \(344 \, \text{cm}^2\)
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Example 2.7.5.

Work out the surface area of the cube whose side is \(8 \, \text{cm}\text{.}\)
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Solution.
The surface area of the cube is:
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The area of one face.
\begin{align*} = \amp 8 \,\text{cm} \times 8 \, \text{cm} \\ = \amp 64 \, \text{cm}^2 \end{align*}
There are 6 faces therefore the surface area of the cube is;
\begin{align*} = \amp 6 \, \times 64 \, \text{cm}^2\\ = \amp 384 \, \text{cm}^2 \end{align*}
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The surface area of the cube is 384 \(\text{cm}^2\)
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Example 2.7.6.

Find the surface area of a cuboid measuring \(8 \, \text{cm}\) by \(3 \, \text{cm}\) by \(14 \, \text{cm}\text{.}\)
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Solution.
The surface area of the cuboid is calculated as:
\begin{equation*} S.A = 2(lw + lh + wh) \end{equation*}
\begin{equation*} = 2((8 \times 3) + (8 \times 14) + (3 \times 14)) \end{equation*}
\begin{equation*} = 2(24 + 112 + 42) \end{equation*}
\begin{equation*} = 2(178) \end{equation*}
\begin{equation*} = 356 \text{ cm}^2 \end{equation*}
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Checkpoint 2.7.7.

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Checkpoint 2.7.8.

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Checkpoint 2.7.9.

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Checkpoint 2.7.10.

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Checkpoint 2.7.11.

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Exercises Exercises

1.

A cube has side length \(9 \, \text{cm}\text{.}\) Find its total surface area.
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Answer.
\(486 \, \text{cm}^2\)
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2.

A company is designing a cube-shaped promotional stand with a side length of \(5\) meters. The stand will be covered with high-quality wallpaper on all six faces. If one roll of wallpaper covers \(10\) square meters, determine the number of rolls required to fully cover the cube.
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3.

A cube-shaped metal box has side \(1.5 \text{m}\text{.}\) If metal sheets cost \(Ksh. 2,000\) per \(\text{m}^2\text{,}\) calculate the total cost of making the box.
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Solution.
A cube with side \(s = 1.5\,\text{m}\) has surface area
\begin{equation*} \text{SA} = 6s^2 = 6\times(1.5)^2 = 6\times2.25 = 13.5 \, \text{m}^2\text{.} \end{equation*}
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Cost at \(2000\,\text{Ksh/m}^2\) is
\begin{equation*} 13.5 \times 2000 = 27{,}000 \, \text{Ksh}\text{.} \end{equation*}
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4.

A rectangular box measures \(12 \, \text{cm}\) by \(7 \, \text{cm}\) by \(5 \, \text{cm}\text{.}\) Calculate its total surface area.
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Answer.
\(382 \, \text{cm}^2\)
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5.

A cuboid-shaped water tank measures \(4 \, \text{m}\) by \(3 \, \text{m}\) by \(2 \, \text{m}\text{.}\) Only the four walls and the base are painted. Find the area painted.
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Answer.
\(52 \, \text{m}^2\)
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6.

A large wooden dice with sides measuring \(12 \, \text{cm}\) each is to be painted on all six faces. If one milliliter of paint covers \(5 \, \text{cm}^2\text{,}\) calculate the total amount of paint required to cover the dice completely.
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Answer.
The total amount of paint required is \(\left(6 \times 12^2 \right) / 5 = 864 / 5 = 172.8 \, \text{ml}\text{.}\)
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7.

A cube has a side length of \(25 \, \text{cm}\text{.}\) If the cost of the cardboard material is Ksh.\(110\) per square centimeter, determine the total cost of making one box. How much would it cost to produce \(500\) such boxes?
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Answer.
The total cost of making one box is \(Ksh. 82,500\text{.}\)
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The total cost of producing \(500\) boxes is \(Ksh. 41,250,000\text{.}\)
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8.

A small storage room is designed with an edge length of \(4\) meters each side. The floor, four walls, and ceiling all need to be tiled. If each tile covers an area of \(0.5\) square meters, determine the total number of tiles required to fully cover the interior of the room.
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Solution.
For a room with edge length \(s = 4\,\text{m}\text{,}\) the total interior area to tile (floor + ceiling + four walls) is the total surface area of a cube:
\begin{equation*} \text{SA} = 6s^2 = 6\times4^2 = 6\times16 = 96 \, \text{m}^2\text{.} \end{equation*}
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Each tile covers \(0.5\,\text{m}^2\text{,}\) so number of tiles required is
\begin{equation*} \dfrac{96}{0.5} = 192 \end{equation*}
tiles.
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9.

A rectangular box measures \(12 \text{cm}\) by \(7 \text{cm}\) by \(5 \text{cm}\text{.}\) Calculate its total surface area.
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Solution.
Total surface area of a rectangular box:
\begin{equation*} \text{SA} = 2(lw + lh + wh)\text{.} \end{equation*}
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Substituting \(l=12\,\text{cm},\; w=7\,\text{cm},\; h=5\,\text{cm}\) gives:
\begin{align*} \text{SA}\amp = 2(12\times7 + 12\times5 + 7\times5) \\ \amp = 2(84 + 60 + 35) \\ \amp = 2(179) = 358 \, \text{cm}^2 \end{align*}
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10.

A rectangular cardboard box has dimensions of \(10 \, \text{cm}\) by \(8 \, \text{cm}\) by \(5 \, \text{cm}\text{.}\) Calculate its total surface area of the material required to construct the box.
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Answer.
\(340 \, \text{cm}^2\text{.}\)
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11.

A pet shop wants to construct a cube-shaped aquarium with a side length of \(1.2\) meters. The aquarium needs to be made entirely of glass, including the base and all four vertical sides, but the top will remain open. If the cost of glass is Ksh. \(750\) per square meter, find the total cost of constructing the aquarium.
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Answer.
\(Ksh. 10,800\)
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12.

  1. A gift shop sells a rectangular gift box with dimensions \(30 \, \text{cm}\) by \(20 \, \text{cm}\) by \(12 \, \text{cm}\text{.}\) If the shop owner wants to wrap the entire box, including all its faces, calculate the minimum amount of wrapping paper needed. If the wrapping paper is sold in rolls of \(1 \, \text{m}^2\text{,}\) how many rolls would be needed to wrap \(50\) boxes?
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  2. A gift box is being wrapped for a special occasion, and it has dimensions of \(20 \, \text{cm}\) in length, \(12 \, \text{cm}\) in width, and \(10 \, \text{cm}\) in height. Calculate the exact amount of wrapping paper required to cover the entire box without any overlap.
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Answer.
  1. \(\displaystyle 12 \, \text{rolls}\)
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  2. \(\displaystyle 1120 \, \text{cm}^2\)
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13.

A metallic shipping box measures \(25 \, \text{cm}\) by \(15 \, \text{cm}\) by \(10 \, \text{cm}\text{.}\) Calculate the total amount of sheet metal required to construct the box, assuming no material is wasted.
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Answer.
\(1300 \, \text{cm}^2\)
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14.

The total surface area of a cuboid is given as \(484 \, \text{cm}^2\) and two of its dimensions are \(8 \, \text{cm}\) and \(6 \, \text{cm}\text{.}\) Determine the missing height (h) of the cuboid.
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Answer.
\(\approx 13.86 \, \text{cm}\)
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15.

The school’s storage room has dimensions of \(4 \, \text{m} \times 5 \, \text{m} \times 3 \, \text{m}\text{.}\) If the walls, floor and ceiling need to be painted, determine the total area that will be covered with paint.
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Answer.
\(94 \, \text{m}^2\)
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16.

A classroom has a length of \(10\) meters, a width of \(8\) meters and a height of \(4\) meters. The four walls and the ceiling need to be painted, but the floor is covered with tiles. If one litre of paint covers \(5\) square meters, calculate the total area to be painted and determine the amount of paint required.
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Answer.
Total area to be painted: \(160 \, \text{m}^2\)
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Amount of paint required: \(32 \, \text{litres}\)
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17.

\(\textbf{Awasii company}\) is designing packaging box with dimensions \(25 \, \text{cm}\) by \(15 \, \text{cm}\) by \(10 \, \text{cm}\text{.}\) The company wants to reduce costs by using the minimum possible material while ensuring the entire box is covered. Calculate the total surface area of the box and determine the cost of producing \(1,000\) such boxes if the material costs Ksh. \(135\) per square centimeter.
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Answer.
Total surface area of the box: \(1300 \, \text{cm}^2\)
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Cost of producing 1,000 boxes: Ksh. \(135,000\)
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18.

A metal storage container is shaped like a cuboid with dimensions \(6 \, \text{m}\) by \(4 \, \text{m}\) by \(3 \, \text{m}\text{.}\) The container needs to be insulated on all its surfaces except for one of the \(6 \, \text{m}\) by \(4 \, \text{m}\) walls, which serves as the entrance. If the insulation material costs Ksh.\(295\) per square meter, determine the total cost of insulating the container.
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Answer.
Total surface area of the container: \(108 \, \text{m}^2\)
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Area to be insulated: \(96 \, \text{m}^2\)
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Total cost of insulating the container: Ksh. \(28,320\)
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19.

A cube-shaped metal water tank with a side length of \(2.5\) meters is being coated with a protective layer on all its surfaces to prevent rusting. If the coating material costs Ksh.\(150\) per square meter, calculate the total cost to coat the entire tank.
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Answer.
Total cost to coat the tank: Ksh. \(5,625\)
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