Samacheer Kalvi Class 9 Maths Solutions Chapter 7 Mensuration Exercise 7.2

Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 7 Mensuration Ex 7.2

Question 1. Find the Total Surface Area and the Lateral Surface Area of a cuboid whose dimensions are length = 20 cm, breadth = 15 cm, height = 8 cm.
Answer: For the given cuboid:
Length \( (l) = 20 \) cm
Breadth \( (b) = 15 \) cm
Height \( (h) = 8 \) cm

First, calculate the Lateral Surface Area (L.S.A.) of the cuboid, which is the area of its four side walls.
L.S.A. of the cuboid \( = 2(l + b)h \) sq. units
\( = 2(20 + 15) \times 8 \)
\( = 2(35) \times 8 \)
\( = 70 \times 8 \)
\( = 560 \) sq. cm

Next, calculate the Total Surface Area (T.S.A.) of the cuboid, which includes all six faces.
T.S.A. of the cuboid \( = 2(lb + bh + lh) \) sq. units
\( = 2(20 \times 15 + 15 \times 8 + 8 \times 20) \)
\( = 2(300 + 120 + 160) \)
\( = 2(580) \)
\( = 1160 \) sq. cm
In simple words: We find the area of the sides (L.S.A.) and then the area of all six faces (T.S.A.) using the given length, breadth, and height of the cuboid.

🎯 Exam Tip: Remember the formulas for L.S.A. \( 2(l+b)h \) and T.S.A. \( 2(lb+bh+lh) \) of a cuboid. Make sure to use consistent units for all dimensions before calculating.

Question 2. The dimensions of a cuboidal box are 6 m x 400 cm x 1.5 m. Find the cost of painting its entire outer surface at the rate of Rs 22 per m².
Answer: The dimensions of the cuboidal box are:
Length \( (l) = 6 \) m
Breadth \( (b) = 400 \) cm \( = 4 \) m (since 100 cm = 1 m, we convert 400 cm to 4 m)
Height \( (h) = 1.5 \) m

To find the cost of painting its entire outer surface, we need to calculate the Total Surface Area (T.S.A.) of the cuboid.
T.S.A. of the cuboid \( = 2(lb + bh + lh) \) sq. units
\( = 2(6 \times 4 + 4 \times 1.5 + 1.5 \times 6) \)
\( = 2(24 + 6 + 9) \)
\( = 2(39) \)
\( = 78 \) sq. m

The cost of painting for one square meter is Rs 22.
Total cost of painting \( = \text{T.S.A.} \times \text{Rate per sq.m} \)
\( = 78 \times 22 \)
\( = \text{Rs } 1716 \)
In simple words: First, we change all measurements to meters. Then, we find the total outside area of the box. Finally, we multiply this area by the cost to paint one square meter to get the total painting cost.

🎯 Exam Tip: Always ensure all dimensions are in the same unit before starting calculations. Convert all measurements to the unit specified for the rate (e.g., meters for cost per m²).

Question 3. The dimensions of a hall is 10 m × 9m×8m. Find the cost of white washing the walls and ceiling at the rate of Rs 8.50 per m².
Answer: The dimensions of the hall are:
Length \( (l) = 10 \) m
Breadth \( (b) = 9 \) m
Height \( (h) = 8 \) m

To whitewash the walls and ceiling, we need to find the area of the four walls (Lateral Surface Area) plus the area of the ceiling. The floor is not whitewashed.
Area to be whitewashed \( = \text{L.S.A. } + \text{ Area of the ceiling} \)
L.S.A. of the hall \( = 2(l + b)h \)
Area of the ceiling \( = lb \)
Area to be whitewashed \( = 2(l + b)h + lb \) sq. units
\( = 2(10 + 9)8 + 10 \times 9 \)
\( = 2(19)8 + 90 \)
\( = 38 \times 8 + 90 \)
\( = 304 + 90 \)
\( = 394 \) sq. m

The cost of whitewashing one square meter is Rs 8.50.
Total cost of whitewashing \( = \text{Area to be whitewashed} \times \text{Rate per sq.m} \)
\( = 394 \times 8.50 \)
\( = \text{Rs } 3349 \)
In simple words: We calculate the area of the four walls and the ceiling of the hall. Then, we multiply this total area by the given cost per square meter to find the overall cost for whitewashing.

🎯 Exam Tip: When calculating areas for painting or whitewashing, always identify which surfaces are included (e.g., walls and ceiling, but not the floor) to apply the correct formula combinations.

Question 4. Find the TSA and LSA of the cube whose side is
(i) 8 m
(ii) 21 cm
(iii) 7.5 cm
Answer:
(i) Side of a cube \( (a) = 8 \) m
Total Surface Area (T.S.A.) of the cube \( = 6a^2 \) sq. units
\( = 6 \times 8 \times 8 \)
\( = 6 \times 64 \)
\( = 384 \) sq. m

Lateral Surface Area (L.S.A.) of the cube \( = 4a^2 \) sq. units
\( = 4 \times 8 \times 8 \)
\( = 4 \times 64 \)
\( = 256 \) sq. m

(ii) Side of a cube \( (a) = 21 \) cm
Total Surface Area (T.S.A.) of the cube \( = 6a^2 \) sq. units
\( = 6 \times 21 \times 21 \)
\( = 6 \times 441 \)
\( = 2646 \) sq. cm

Lateral Surface Area (L.S.A.) of the cube \( = 4a^2 \) sq. units
\( = 4 \times 21 \times 21 \)
\( = 4 \times 441 \)
\( = 1764 \) sq. cm

(iii) Side of a cube \( (a) = 7.5 \) cm
Total Surface Area (T.S.A.) of the cube \( = 6a^2 \) sq. units
\( = 6 \times 7.5 \times 7.5 \)
\( = 6 \times 56.25 \)
\( = 337.5 \) sq. cm

Lateral Surface Area (L.S.A.) of the cube \( = 4a^2 \) sq. units
\( = 4 \times 7.5 \times 7.5 \)
\( = 4 \times 56.25 \)
\( = 225 \) sq. cm
In simple words: For each cube, we use the side length to calculate its total surface area (area of all six faces) and its lateral surface area (area of the four side faces only).

🎯 Exam Tip: Remember that for a cube, T.S.A. is \( 6a^2 \) and L.S.A. is \( 4a^2 \). Be careful with units (m or cm) and decimal calculations.

Question 5. If the total surface area of a cube is 2400 cm² then, find its lateral surface area.
Answer: We are given that the Total Surface Area (T.S.A.) of a cube is 2400 cm².
The formula for T.S.A. of a cube is \( 6a^2 \), where 'a' is the side length.
So, \( 6a^2 = 2400 \)
To find the side length, we divide by 6:
\( a^2 = \frac{2400}{6} \)
\( a^2 = 400 \)
Now, we take the square root of both sides to find 'a':
\( a = \sqrt{400} \)
\( a = 20 \) cm

Once we have the side length, we can find the Lateral Surface Area (L.S.A.) of the cube.
The formula for L.S.A. of a cube is \( 4a^2 \).
L.S.A. of the cube \( = 4 \times (20)^2 \)
\( = 4 \times 400 \)
\( = 1600 \) cm²
In simple words: First, we use the given total surface area to figure out the length of one side of the cube. Then, using that side length, we calculate the area of only the four side faces, which is the lateral surface area.

🎯 Exam Tip: When given the total surface area, always find the side length first using the T.S.A. formula before calculating the lateral surface area or any other dimension.

Question 6. A cubical container of side 6.5 m is to be painted on the entire outer surface. Find the area to be painted and the total cost of painting it at the rate of Rs 24 per m².
Answer: The side of the cubical container \( (a) = 6.5 \) m.

Since the container is to be painted on its entire outer surface, we need to calculate its Total Surface Area (T.S.A.).
T.S.A. of the cube \( = 6a^2 \) sq. units
\( = 6 \times (6.5)^2 \)
\( = 6 \times 6.5 \times 6.5 \)
\( = 6 \times 42.25 \)
\( = 253.50 \) sq. m
So, the area to be painted is 253.50 sq. m.

The cost of painting for one square meter is Rs 24.
Total cost of painting \( = \text{Area to be painted} \times \text{Rate per sq.m} \)
\( = 253.50 \times 24 \)
\( = \text{Rs } 6084 \)
In simple words: We find the total outside area of the cube using its side length. Then, we multiply this total area by the cost to paint each square meter to find the full cost of painting.

🎯 Exam Tip: Always make sure to square the side length correctly when calculating \( a^2 \), especially with decimals. Double-check your multiplication for the total cost.

Question 7. Three identical cubes of side 4 cm are joined end to end. Find the total surface area and lateral surface area of the new resulting cuboid.
Answer: When three identical cubes, each with a side of 4 cm, are joined end to end, they form a new cuboid. The dimensions of this new cuboid will change as follows:

Length of the new cuboid \( (l) = \text{side of cube} + \text{side of cube} + \text{side of cube} \)
\( = 4 + 4 + 4 \)
\( = 12 \) cm

Breadth of the new cuboid \( (b) = \text{side of cube} \)
\( = 4 \) cm

Height of the new cuboid \( (h) = \text{side of cube} \)
\( = 4 \) cm

Now we calculate the Total Surface Area (T.S.A.) of the new cuboid:
T.S.A. of the new cuboid \( = 2(lb + bh + lh) \) sq. units
\( = 2(12 \times 4 + 4 \times 4 + 4 \times 12) \)
\( = 2(48 + 16 + 48) \)
\( = 2(112) \)
\( = 224 \) cm²

Next, we calculate the Lateral Surface Area (L.S.A.) of the new cuboid:
L.S.A. of the new cuboid \( = 2(l + b)h \) sq. units
\( = 2(12 + 4) \times 4 \)
\( = 2(16) \times 4 \)
\( = 32 \times 4 \)
\( = 128 \) cm²
In simple words: When three cubes are put together in a line, they form a longer shape. We find its new length, while its width and height stay the same as one cube. Then, we use these new measurements to find the total outer area and the area of its four sides.

🎯 Exam Tip: Visualize how joining the cubes affects the dimensions. Only the length changes when cubes are joined end-to-end; breadth and height remain the same as the original cube's side.