RBSE Solutions Class 9 Maths Chapter 12 Surface Area and Volume of Cube and Cuboid Exercise 12.1

Question 1. The length, breadth and height of a closed wooden box are 1 m, 60 cm and 40 cm respectively. Find the external surface area of the box.
Answer: First, we convert all dimensions to the same unit, centimetres.
Length (l) = 1 m = 100 cm
Breadth (b) = 60 cm
Height (h) = 40 cm
The external surface area of a closed wooden box (cuboid) is calculated using the formula:
External surface area \( = 2(lb + bh + hl) \)
Now, we put in the values:
\( = 2(100 \times 60 + 60 \times 40 + 40 \times 100) \)
\( = 2(6000 + 2400 + 4000) \)
\( = 2(12400) \)
\( = 24800 \, cm^2 \)
To convert this to square metres, we remember that \( 1 \, m^2 = 100 \times 100 \, cm^2 = 10000 \, cm^2 \).
\( = \frac {24800}{10000} \, m^2 \)
\( = 2.48 \, m^2 \)
In simple words: We find the total area of all sides of the box. First, change all measurements to centimetres. Then, use the formula for the surface area of a box and calculate the total area, converting it to square metres at the end.

🎯 Exam Tip: Always ensure all dimensions are in the same unit before performing calculations. Convert to the required final unit at the end, often metres for area or volume.

Question 2. How many square centimetre will be required to cover the box whose dimensions are 40 cm, 30 cm and 20 cm respectively?
Answer: The problem asks for the total surface area required to cover the box. This is the sum of the areas of all its faces.
Given dimensions of the box:
Length (l) = 40 cm
Breadth (b) = 30 cm
Height (h) = 20 cm
The formula for the total surface area (x) of a cuboid is:
\( x = 2(lb + bh + hl) \)
Now, we substitute the given values into the formula:
\( x = 2(40 \times 30 + 30 \times 20 + 20 \times 40) \)
\( x = 2(1200 + 600 + 800) \)
\( x = 2(2600) \)
\( x = 5200 \, cm^2 \)
Thus, \( 5200 \, cm^2 \) area will be required to cover the whole box. The surface area shows how much material is needed to wrap the entire object.
In simple words: To find out how much material is needed to cover the box, we calculate its total surface area using the given length, breadth, and height. The final answer is \( 5200 \, cm^2 \).

🎯 Exam Tip: Remember that "to cover" a box usually means finding its total surface area. Pay attention to whether the box is open or closed, as this affects which surfaces are included in the calculation.

Question 3. A room is of length 5 metres, breadth 3.5 metres and height 4 metres. Find the cost of four walls and the roof at the rate of Rs. 15 per square metre.
Answer: We need to find the area of the four walls and the roof of the room to calculate the painting cost.
Given dimensions:
Length (l) = 5 metres
Breadth (b) = 3.5 metres
Height (h) = 4 metres
Area of the four walls \( = 2(l+b)h \)
\( = 2(5 + 3.5)4 \)
\( = 2(8.5)4 \)
\( = 17 \times 4 \)
\( = 68 \, m^2 \)
Area of the roof \( = l \times b \)
\( = 5 \times 3.5 \)
\( = 17.5 \, m^2 \)
Total area to be painted \( = \) Area of four walls + Area of roof
\( = 68 + 17.5 \)
\( = 85.5 \, m^2 \)
Cost of painting per square metre = Rs. 15
Total cost of painting \( = \) Total area \( \times \) Rate per square metre
\( = 85.5 \times 15 \)
\( = \text{Rs. } 1282.5 \)
This calculation includes all surfaces that would typically be painted in a room, excluding the floor.
In simple words: We first find the area of the four walls and the ceiling (roof) of the room. We add these areas together to get the total area to be painted. Then, we multiply this total area by the cost of painting for each square metre.

🎯 Exam Tip: For problems involving painting a room, always remember to calculate the area of the four walls and the ceiling, but exclude the floor unless specifically asked. Use the correct formulas for each part.

Question 4. The edge of a cubical chalk box is 4 cm, then find the total surface area of the chalk box and also find the length of its diagonal.
Answer: Given that the edge (side) of the cubical chalk box is \( a = 4 \, cm \).
To find the total surface area of a cube, we use the formula:
Surface Area \( = 6a^2 \)
Substitute the value of 'a':
\( = 6 \times (4)^2 \)
\( = 6 \times 16 \)
\( = 96 \, cm^2 \)
To find the length of the diagonal of a cube, we use the formula:
Diagonal \( = \sqrt{3}a \)
Substitute the value of 'a':
\( = \sqrt{3} \times 4 \)
\( = 4\sqrt{3} \, cm \)
A cube's diagonal connects opposite corners, running through its interior.
In simple words: For a cube with a side of 4 cm, its total outer area is found by multiplying 6 by the square of the side length. The length of its main diagonal is found by multiplying the side length by the square root of 3.

🎯 Exam Tip: Remember the specific formulas for a cube's surface area \( (6a^2) \) and its main diagonal \( (\sqrt{3}a) \). These are fundamental for cube-related problems.

Question 5. The total surface area of a cube is 1014 sq. m. Find the length of its edge.
Answer: Given that the total surface area of a cube is \( 1014 \, m^2 \).
The formula for the total surface area of a cube with edge length 's' is:
Total surface area \( = 6s^2 \)
We can set up the equation using the given information:
\( 1014 = 6s^2 \)
Now, we solve for \( s^2 \):
\( s^2 = \frac{1014}{6} \)
\( s^2 = 169 \)
To find 's', we take the square root of 169:
\( s = \sqrt{169} \)
\( s = 13 \, m \)
So, the length of the edge of the cube is 13 metres. Each side of the cube would be 13 metres long.
In simple words: We know the total surface area of a cube. We use the formula for a cube's surface area, divide the given area by 6, and then find the square root of that number to get the length of one side (edge).

🎯 Exam Tip: When given the total surface area of a cube and asked to find its edge, simply divide the area by 6 (since there are 6 equal faces) and then take the square root of the result.

Question 6. The inner dimensions of a closed wooden box are 1 metre, 65 cm and 55 cm. Thickness of the wood is 2.5 cm. Find the cost of painting the outer surface at the rate of Rs. 15 per square metre.
Answer: First, let's list the inner dimensions and the wood thickness, ensuring consistent units (centimetres).
Inner length (l) = 1 m = 100 cm
Inner breadth (b) = 65 cm
Inner height (h) = 55 cm
Thickness of the wood = 2.5 cm
Since it's a closed box, the wood adds thickness to both sides (length, breadth, and height).
Outer length (L) = Inner length + \( 2 \times \) Thickness
\( L = 100 + 2 \times 2.5 = 100 + 5 = 105 \, cm = 1.05 \, m \)
Outer breadth (B) = Inner breadth + \( 2 \times \) Thickness
\( B = 65 + 2 \times 2.5 = 65 + 5 = 70 \, cm = 0.70 \, m \)
Outer height (H) = Inner height + \( 2 \times \) Thickness
\( H = 55 + 2 \times 2.5 = 55 + 5 = 60 \, cm = 0.60 \, m \)
Now, we calculate the outer surface area of the box using the outer dimensions.
Outer Surface Area \( = 2(LB + BH + HL) \)
\( = 2(1.05 \times 0.70 + 0.70 \times 0.60 + 0.60 \times 1.05) \)
\( = 2(0.735 + 0.42 + 0.63) \)
\( = 2(1.785) \)
\( = 3.570 \, m^2 \)
The cost of painting is Rs. 15 per square metre.
Total cost of painting \( = \) Outer Surface Area \( \times \) Rate per square metre
\( = 3.570 \times 15 \)
\( = \text{Rs. } 53.55 \)
Therefore, the cost to paint the outer surface of the wooden box is Rs. 53.55.
In simple words: First, we find the outside measurements of the box by adding the wood's thickness to its inside measurements. Then, we use these outside measurements to calculate the total outer area of the box. Finally, we multiply this total area by the painting cost per square metre to get the total painting cost.

🎯 Exam Tip: Be careful when calculating outer dimensions for a closed box: thickness is added to *both* sides of each dimension. Also, convert all units to metres before calculating cost if the rate is per square metre.

Question 7. Each surface of a cube is 100 square cm. If it is cut parallel to the base by a plane in two equal parts. Find the total surface area of each part.
Answer: Given that each surface (face) of the cube has an area of \( 100 \, cm^2 \).
When a cube is cut parallel to its base into two equal parts, it forms two smaller cuboids. Let's analyze the surface area change for one of these new parts:
1. **Original top/bottom face:** Each new part will have one original face (either the top or the bottom of the original cube), which has an area of \( 100 \, cm^2 \).
2. **New cut face:** The plane cut creates two new surfaces. Each new part will have one of these surfaces, which will have the same area as the original cube's face, i.e., \( 100 \, cm^2 \).
3. **Side faces:** The original four side faces of the cube are now split in half. So, each of these four side faces will contribute half its area to each new part. Half of \( 100 \, cm^2 \) is \( 50 \, cm^2 \). There are four such faces for each part.
Therefore, the total surface area of each separated part will be:
Area of each part \( = \) (Area of original top/bottom face) + (Area of new cut face) + (Area of 4 halved side faces)
\( = 100 \, cm^2 + 100 \, cm^2 + (50 + 50 + 50 + 50) \, cm^2 \)
\( = 100 \, cm^2 + 100 \, cm^2 + 200 \, cm^2 \)
\( = 400 \, cm^2 \)
So, the total surface area of each separated portion is \( 400 \, cm^2 \). This method shows how the surface area changes when an object is divided.
In simple words: When a cube is cut in half horizontally, each new piece has its original top or bottom face, plus a new flat face from the cut. The four side faces are also split in half. We add the area of the original face, the new cut face, and the four half-side faces to find the total area of each new part.

🎯 Exam Tip: When an object is cut, new surfaces are created, which add to the total surface area. Always identify the original faces that remain, the new faces created by the cut, and any faces that are divided.

Question 8. An open box is made of wood 3 cm thick. Its external length, breadth and height are 146 cm, 116 cm and 83 cm. Find the inside surface area.
Answer: We need to find the internal dimensions first, considering the wood's thickness and that the box is open (no top).
Given:
External length (L) = 146 cm
External breadth (B) = 116 cm
External height (H) = 83 cm
Thickness of the wood = 3 cm
For an open box, the thickness affects length and breadth from both sides (left/right, front/back), but height only from the bottom (as there is no top).
Inner length (l) = External length \( - 2 \times \) Thickness
\( l = 146 - (2 \times 3) = 146 - 6 = 140 \, cm \)
Inner breadth (b) = External breadth \( - 2 \times \) Thickness
\( b = 116 - (2 \times 3) = 116 - 6 = 110 \, cm \)
Inner height (h) = External height \( - 1 \times \) Thickness (since it's open at the top)
\( h = 83 - (1 \times 3) = 83 - 3 = 80 \, cm \)
Now, we calculate the inside surface area, which includes the area of the four inner walls and the inner base (since it's an open box, there's no inner top).
Area of four inner walls \( = 2(l+b)h \)
\( = 2(140 + 110) \times 80 \)
\( = 2(250) \times 80 \)
\( = 500 \times 80 \)
\( = 40,000 \, cm^2 \)
Area of inner base \( = l \times b \)
\( = 140 \times 110 \)
\( = 15,400 \, cm^2 \)
Total inside surface area \( = \) Area of four inner walls + Area of inner base
\( = 40,000 + 15,400 \)
\( = 55,400 \, cm^2 \)
This total area represents the surface inside the box that could be painted or lined.
In simple words: First, we subtract the wood's thickness from the external measurements to find the internal length, breadth, and height. Remember to only subtract thickness once from the height because the box is open. Then, we calculate the area of the four inner walls and the inner bottom (base) and add them together to get the total inside surface area.

🎯 Exam Tip: For an open box, be careful when calculating inner dimensions: thickness is subtracted twice for length and breadth, but only once for height (from the base). The inner surface area excludes the top face.