In this chapter we will discuss total and lateral surface area of cube questions with solutions.
The formula for both the calculation is given as;
Surface area of cube = \mathtt{6a^{2}}
Lateral surface area of cube = \mathtt{6=4a^{2}}
The total surface area is the sum of area of all the square walls that form the cube.
The lateral surface area is the sum of area of side walls (excluding roof and bottom)
Cube surface area – solved problems
Question 01
Find the lateral and total surface area of cube of length 12 cm.
Solution
Lateral surface area = \mathtt{4a^{2}}\\ \\
Lateral surface area = \mathtt{4\ ( 12)^{2} =\ 576\ cm^{2}}
Total surface area = \mathtt{6a^{2}}
Total surface area = \mathtt{6\ ( 12)^{2} =\ 864\ cm^{2}}
Question 02
Joey bought a beef meat in the form of cube of length 6 cm. She then cuts the meat into smaller cubes of 1 cm length for serving purpose. Calculate the total surface area of all the small cubes combined.
Solution
The number of smaller cubes after cutting big cube is given by following formula;
Number of small cube = \mathtt{\left(\frac{x}{y}\right)^{3}}
Where;
x = length of big cube
y = length of small cube
Here, the big cube of 6 units is cut into cube of 1 cm.
Number of small cube = \mathtt{( 6)^{3}} = 216
Hence, there will be 216 cubes of 1 cm each.
Total surface area of 1 cm cube = \mathtt{6\times ( 1)^{2}} = 6 sq cm.
Total surface area of 216 cubes = 216 x 6 = 1296 sq. cm
Question 03
Rashid bought a cubical aquarium of length 1 meter. He wants to cover two opposite side walls with colored paper. If the cost of colored paper is $3 /sq m, find the total expense incurred.
Solution
Area of one side = 1 x 1 = 1 sq. meter.
Since two opposite walls are covered, the total area will be = 1 + 1 = 2 sq. meter
Now let’s calculate the cost.
1 sq meter cost = 3 dollars
2 sq meter cost = 3 x 2 = 6 dollars
Hence, total of 6 dollars will be incurred to cover the two walls.
Question 04
A blacksmith decided to create a hollow cubical box of length 45 cm. If the cost of iron sheet is 0.001$ per square cm. Find the cost to build the hollow cube.
Solution
Calculate the total surface area.
Total surface area = [/latex] \mathtt{=\ 6\times ( 45)^{2} =\ 12150\ cm^{2}} [/latex]
Hence, total of 12150 sq cm of iron sheet is required to build the cube.
Total cost incurred will be;
1 sq cm cost = 0.001 sq. cm
12150 sq cm cost = 12150 x 0.001= $ 12.15
Hence, 12.15 is the incurred cost.
Question 05
An open cubical box is made of wood 5 cm thick. If the external side of the cube is 30 cm. Find the cost to paint the inner surface at the rate .005$ per sq cm. Note that the cube is open from the top.
Solution
Fine the dimension of inner cube.
Internal length = 30 -2(5) = 20 cm
Internal width = 30 -2(5) = 20 cm
Internal height = 30 – 5 = 25 cm (top is open)
Surface area of open cube = 2(lh + bh) +lb
\mathtt{\Longrightarrow \ 2\ ( 20.25+20.25) +\ ( 20.20)}\\\ \\ \mathtt{\Longrightarrow \ 2\ ( 500\ +\ 500) \ +\ 400}\\\ \\ \mathtt{\Longrightarrow \ 2000\ +\ 400}\\\ \\ \mathtt{\Longrightarrow \ 2400\ cm^{2}}
So, 2400 sq cm of area will be painted.
Now let’s calculate the total cost.
.005$ per sq cm
1 sq cm ⟹ .005$
2400 sq cm ⟹ 2400 x 0.005 = 12 $
Hence, total cost to paint will be 12$