Class 9 Mathematics Exercise-11.1 NCERT Solutions Olympiad Board Exam

Chapter 11

SURFACE AREAS AND VOLUMES

Step-by-step NCERT solutions with stress–strain analysis and exam-oriented hints for Boards, JEE & NEET.

8 Questions

20–25 min Ideal time

Q1 Now at

📘 Concept & Theory Concept Used ›

A cone is a three-dimensional solid having a circular base and a curved surface meeting at a single point called the vertex.

The curved surface area of a cone represents the area covered only by the curved portion of the cone, excluding the circular base.

The formula for curved surface area of a cone is:

\[\small \text{Curved Surface Area of Cone} = \pi r l \]

where,

\[\small r = \text{radius of the base} \]
\[\small l = \text{slant height of the cone} \]
\[\small \pi = \frac{22}{7} \] (used when numerical values are convenient)

🗺️ Solution Roadmap Step-by-step Plan ›

Find the radius using the given diameter.
Use the formula \[\small \pi r l \] for curved surface area.
Substitute the values carefully.
Simplify step by step to obtain the final answer.

📊 Graph / Figure Graph / Figure ›

Cone showing radius and slant height

✏️ Solution Complete Solution ›

Step-by-step Solution · 9 steps

Diameter of the base of the cone is given as:\[\small d = 10.5~\text{cm}\]
Radius is half of the diameter.\[\small r = \frac{d}{2}\]
Substituting the value of diameter:\[\small \begin{aligned}r &= \frac{10.5}{2}\\r &= 5.25~\text{cm}\end{aligned}\]
Slant height of the cone is given as:\[\small l = 10~\text{cm}\]
Formula for curved surface area of a cone:\[\small \text{Curved Surface Area} = \pi r l\]
Substituting the known values:\[\small = \frac{22}{7} \times 5.25 \times 10\]
Convert decimal into multiplication form for easy simplification:\[\small 5.25 = \frac{525}{100}\]
Therefore,\[\small = \frac{22}{7} \times \frac{525}{100} \times 10\]
Simplifying\[\small \begin{aligned}&= \frac{22 \times 525 \times 10}{7 \times 100}\\ &= \frac{22 \times 75 \times 10}{100}\\ &= \frac{16500}{100}\\ &= 165\end{aligned}\]

💡 Answer Final Answer ›

Answer: The curved surface area of the cone is: \(\small 165~\text{cm}^{2}\)

🎯 Exam Significance Exam Significance ›

This question strengthens the understanding of the formula for curved surface area of a cone.
Questions based on surface areas are frequently asked in school examinations, unit tests, and board examinations.
Competitive examinations often test direct formula application along with correct unit handling and simplification.
It develops accuracy in substituting numerical values into mensuration formulas.
Understanding such basic problems is essential before solving advanced problems involving combinations of solids.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Radius is always half of the diameter.
Curved surface area of a cone is calculated using: \[\small \pi r l \]
Slant height is used in curved surface area calculations, not vertical height.
Proper simplification step by step reduces calculation mistakes.
Final answers for surface area must be written in square units.

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1 / 8 · 13%

Q2 →

📘 Concept & Theory Concept Used ›

The total surface area of a cone includes:

Curved surface area of the cone
Area of the circular base

Therefore, the formula for total surface area of a cone is:

\[\small \text{Total Surface Area of Cone} = \pi r(r+l) \]

where,

\[\small r = \text{radius of the base} \]
\[\small l = \text{slant height of the cone} \]
\[\small \pi = \frac{22}{7} \]

🗺️ Solution Roadmap Step-by-step Plan ›

Find the radius from the given diameter.
Write the formula for total surface area of a cone.
Substitute the values of radius and slant height.
Simplify carefully step by step.
Write the answer with correct square units.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Complete Solution ›

Step-by-step Solution · 11 steps

Slant height of the cone is given as:\[\small l = 21~\text{m}\]
Diameter of the base is:\[\small d = 24~\text{m}\]
Radius is half of the diameter.\[\small r = \frac{d}{2}\]
Substituting the value of diameter: \[\small \begin{aligned}r &= \frac{24}{2}\\ &r = 12~\text{m}\end{aligned}\]
Formula for total surface area of a cone:\[\small \text{TSA} = \pi r(r+l)\]
Substituting the values:\[\small = \frac{22}{7} \times 12 \times (12+21)\]
First simplify the bracket:\[\small = \frac{22}{7} \times 12 \times 33\]
Multiply 12 and 33:\[\small = \frac{22}{7} \times 396\]
Multiply the numerator:\[\small \begin{aligned} &= \frac{22 \times 396}{7}\\&= \frac{8712}{7}\end{aligned}\]
Dividing\[\small = 1244.571428\ldots\]
Approximating to two decimal places:\[\small 1244.57~\text{m}^{2}\]

💡 Answer Final Answer ›

Answer: \(\small \text{TSA }=1244.57~\text{m}^{2}\)

🎯 Exam Significance Exam Significance ›

This problem develops understanding of the difference between curved surface area and total surface area.
Board examinations frequently include direct formula-based mensuration problems.
Competitive entrance examinations test accuracy in formula application and numerical simplification.
Such questions strengthen conceptual clarity regarding radius, diameter, and slant height.
This concept is also useful in practical applications involving tents, funnels, ice-cream cones, and conical containers.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Radius is half of the diameter.
Total surface area of a cone is: \[\small \pi r(r+l) \]
Total surface area includes both curved surface and base area.
Always simplify expressions inside brackets first.
Final answers for surface area must be expressed in square units.

← Q1

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Q3 →

📘 Concept & Theory concept Used ›

The curved surface area of a cone is given by:

\[\small \text{CSA} = \pi r l \]

where,

\[\small r = \text{radius of the base} \]
\[\small l = \text{slant height} \]

After finding the radius, total surface area of the cone can be calculated using:

\[\small \text{TSA} = \pi r(r+l) \]

Total surface area includes:

Curved surface area of the cone
Area of the circular base

🗺️ Solution Roadmap Step-by-step Plan ›

Use the curved surface area formula to find the radius.
Substitute the known values carefully.
Simplify step by step to calculate the radius.
Use the radius obtained to calculate total surface area.
Write the final answers with proper units.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Complete Solution ›

Step-by-step Solution · 7 steps

Curved surface area of the cone is given as:\[\small \text{CSA} = 308~\text{cm}^{2}\]
Slant height is:\[\small l = 14~\text{cm}\]
We know that:\[\small \text{CSA} = \pi r l\]
Substituting the known values:\[\small 308 = \frac{22}{7} \times r \times 14\]
Simplifying: \[\small \begin{aligned} 308 &= \frac{22 \times r \times 14}{7}\\ 308 &= 22 \times r \times 2\\ 308 &= 44r\\ \Rightarrow 44r&=308\\ r&=\frac{308}{44}\\ &=7~\text{cm} \end{aligned} \]
Now, to find the total surface area of the cone:\[\small \text{TSA} = \pi r(r+l)\]
Substituting the values: \[\small \begin{aligned} \text{TSA} &=\frac{22}{7} \times 7 \times (7+14)\\ &=\frac{22}{7} \times 7 \times 21\\ &=22 \times 21\\ &=462\text{ cm}^{2} \end{aligned} \]

💡 Answer Final Answer ›

Answers:
The radius of the base is = \(\small 7~\text{cm}\) and
the total surface area of the cone is: = \(\small 462\text{ cm}^{2}\)

🎯 Exam Significance Exam Significance ›

This question teaches reverse application of mensuration formulas.
Students learn how to find an unknown dimension using surface area formulas.
Such problems are frequently asked in school examinations and board exams.
Competitive entrance exams often include multi-step mensuration problems like this.
It improves algebraic manipulation and numerical simplification skills.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Curved surface area of a cone is: \[\small \pi r l \]
Total surface area of a cone is: \[\small \pi r(r+l) \]
Unknown dimensions can be found by rearranging formulas.
Step-by-step simplification avoids arithmetic mistakes.
Surface areas are always written in square units.

← Q2

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Q4 →

📘 Concept & Theory Concept Used ›

A conical tent has only a curved surface and no base covering. Therefore, only the curved surface area is considered while calculating the amount of canvas required.

The slant height of a cone is calculated using the Pythagoras theorem:

\[\small l^2 = r^2 + h^2 \]

where,

\[\small l = \text{slant height} \]
\[\small r = \text{radius of the base} \]
\[\small h = \text{vertical height} \]

Curved surface area of a cone is:

\[\small \text{CSA} = \pi r l \]

Cost of canvas required:

\[\small \text{Cost} = \text{Area} \times \text{Rate per unit area} \]

🗺️ Solution Roadmap Step-by-step Plan ›

Use the Pythagoras theorem to calculate slant height.
Find the curved surface area of the tent.
Multiply the curved surface area by the cost per square metre.
Write the final answers with proper units.

📊 Graph / Figure Graph / Figure ›

Fig. 1 — Free body diagram

✏️ Solution Complete Solution ›

Step-by-step Solution · 12 steps

Height of the conical tent is:\[\small h = 10~\text{m}\]
Radius of the base is:\[\small r = 24~\text{m}\]
Find the slant height
Using the relation:\[\small l^2 = r^2 + h^2\]
Substituting the values:\[\small l^2 = 24^2 + 10^2\]
Calculating the squares:\[\small \begin{aligned}l^2 &= 576 + 100\\l^2 &= 676\end{aligned}\]
Taking square root on both sides:\[\small \begin{aligned}l &= \sqrt{676}\\&= 26~\text{m}\end{aligned}\]
Therefore, the slant height of the tent is:\[\small 26~\text{m}\]
Find the cost of the canvas required
Since the tent is open at the bottom, only curved surface area is required.
Formula for curved surface area:\[\small \text{CSA} = \pi r l\]
Substituting the values: \[\small \begin{aligned} \text{CSA}&= \frac{22}{7} \times 24 \times 26\\ &=\frac{22}{7} \times 624\\ &= \frac{13728}{7}\\ &= 1961.14~\text{m}^{2} \end{aligned} \]
Cost of canvas for \(\small 1~\text{m}^{2}\) is ₹70.
Therefore, \[\small \begin{aligned}\text{Cost} &= 1961.14 \times 70\\&= 137279.8\end{aligned}\]

💡 Answer Final Answer ›

Answer:
Cost of canvas = \(\small \approx \text{₹ }137280\)

🎯 Exam Significance Exam Significance ›

This problem combines geometry, mensuration, and commercial mathematics.
Students learn practical application of surface area concepts.
Questions involving tents, canvas, and coverings are very common in board examinations.
Competitive exams often include multi-step word problems based on cones.
It strengthens understanding of the Pythagoras theorem in three-dimensional geometry.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Slant height of a cone is calculated using: \[\small l^2 = r^2 + h^2 \]
Canvas required for a tent equals the curved surface area only.
Curved surface area of a cone is: \[\small \pi r l \]
Real-life mensuration problems frequently involve unit cost calculations.
Always write area in square units and cost in currency units.

← Q3

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Q5 →

📘 Concept & Theory Concept Used ›

A conical tent requires only curved surface covering because the base remains open.

To calculate the tarpaulin required, we first find the curved surface area of the cone.

Slant height of the cone is obtained using the Pythagoras theorem:

\[\small l^2 = h^2 + r^2 \]

Curved surface area of a cone is:

\[\small \text{CSA} = \pi r l \]

Since the tarpaulin has fixed width, its required length is:

\[\small \text{Length} = \frac{\text{Area}}{\text{Width}} \]

Extra material is also added for stitching margins and wastage.

🗺️ Solution Roadmap Step-by-step Plan ›

Find the slant height using height and radius.
Calculate the curved surface area of the tent.
Divide the area by the width of tarpaulin to get the required length.
Add extra length for stitching and wastage.
Write the final answer with proper unit.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Complete Solution ›

Step-by-step Solution · 9 steps

Given
Height of the conical tent is:\[\small h = 8~\text{m}\]
Radius of the base is:\[\small r = 6~\text{m}\]
Width of tarpaulin is:\[\small 3~\text{m}\]
Find the slant height
Using the relation:\[\small l^2 = h^2 + r^2\]
Substituting the values: \[\small \begin{aligned} l^2 &= 8^2 + 6^2\\ &= 64 + 36\\ &= 100\\ \Rightarrow l&= \sqrt{100}\\ &= 10~\text{m} \end{aligned} \]
Find the curved surface area of the tent
Formula for curved surface area: \[\small \begin{aligned} \text{CSA} &= \pi r l\\ &= 3.14 \times 6 \times 10\\ &= 18.84 \times 10\\ &= 188.4~\text{m}^{2} \end{aligned} \]
Find the required length of tarpaulin
Since the tarpaulin is 3 m wide: \[\small \begin{aligned} \text{Length} &= \frac{\text{Area}}{\text{Width}}\\ &= \frac{188.4}{3}\\ &= 62.8~\text{m} \end{aligned} \]
Add extra material for stitching and wastage
Extra length required: \[\small \begin{aligned} 20~\text{cm} &= \frac{20}{100}~\text{m}\\ &= 0.2~\text{m} \end{aligned} \]
Therefore,\[\small \begin{aligned}\text{Total length} &= 62.8 + 0.2\\&= 63~\text{m}\end{aligned}\]

💡 Answer Final Answer ›

Answer: Therefore, the required length of tarpaulin is: \(\small 63~\text{m}\)

🎯 Exam Significance Exam Significance ›

This problem demonstrates practical application of surface area in real life.
Students learn how geometry is used in manufacturing and material estimation.
Questions based on tents, coverings, and tarpaulin are commonly asked in board examinations.
Competitive entrance exams often include multi-concept mensuration problems.
The question improves understanding of unit conversion and estimation techniques.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Slant height is found using: \[\small l^2 = h^2 + r^2 \]
Curved surface area of a cone is: \[\small \pi r l \]
Tarpaulin required depends on area and width of the material.
Extra material must always be considered for stitching and wastage.
Proper unit conversion is important in practical mensuration problems.

← Q4

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Q6 →

📘 Concept & Theory Concept Used ›

White-washing the conical tomb means covering only its curved outer surface.

Therefore, we calculate the curved surface area of the cone.

Formula for curved surface area of a cone:

\[\small \text{CSA} = \pi r l \]

where,

\[\small r = \text{radius of the base} \]
\[\small l = \text{slant height} \]

The cost is calculated using:

\[\small \text{Cost} = \frac{\text{Rate} \times \text{Required Area}}{100} \]

🗺️ Solution Roadmap Step-by-step Plan ›

Find the radius using the given diameter.
Calculate the curved surface area of the cone.
Use the given white-washing rate to calculate the total cost.
Write the final answer with correct units.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Complete Solution ›

Step-by-step Solution · 8 steps

Given
Slant height of the conical tomb is:\[\small l = 25~\text{m}\]
Diameter of the base is:\[\small d = 14~\text{m}\]
Radius is half of the diameter.\[\small \begin{aligned}r &= \frac{d}{2}\\&= \frac{14}{2}\\&=7\end{aligned}\]
Find the curved surface area
Formula for curved surface area:\[\small \text{CSA} = \pi r l\]
Substituting the values: \[\small \begin{aligned} \text{CSA} &=\frac{22}{7} \times 7 \times 25\\ &=22 \times 25\\ &= 550\text{ m}^{2} \end{aligned} \]
Find the cost of white-washing
Cost of white-washing \(\small 100~\text{m}^{2}\) is ₹210.
Therefore, cost per \(\small 1~\text{m}^{2}\) is:\[\small \frac{210}{100}=2.1\]
Hence, cost for \(\small 550~\text{m}^{2}\):\[\small \begin{aligned}\text{Cost} &= 550 \times 2.1\\&= 1155\end{aligned}\]

💡 Answer Final Answer ›

Answer: The cost of white-washing the curved surface is: \(\small \text{₹}1155\)

🎯 Exam Significance Exam Significance ›

This question combines mensuration with real-life commercial applications.
Students learn how surface area concepts are used in painting and white-washing problems.
Such practical geometry questions are frequently asked in board examinations.
Competitive entrance examinations often include area and cost estimation problems.
It strengthens calculation accuracy and unit handling skills.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Radius is always half of the diameter.
Curved surface area of a cone is: \[\small \pi r l \]
White-washing or painting problems usually involve only exposed surfaces.
Cost problems require multiplication of area and rate.
Final answers should include both numerical value and proper unit/currency.

← Q5

6 / 8 · 75%

Q7 →

📘 Concept & Theory Concept Used ›

A joker’s cap is shaped like a cone and is open at the bottom. Therefore, only the curved surface area is required to make the cap.

Slant height of the cone is found using the Pythagoras theorem:

\[\small l^2 = r^2 + h^2 \]

Curved surface area of a cone is:

\[\small \text{CSA} = \pi r l \]

To find the sheet required for multiple caps:

\[\small \text{Total Area} = \text{Area of one cap} \times \text{Number of caps} \]

🗺️ Solution Roadmap Step-by-step Plan ›

Calculate the slant height using radius and height.
Find the curved surface area of one cap.
Multiply by 10 to get the total sheet area required.
Write the final answer with correct square units.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Complete Solution ›

Step-by-step Solution · 10 steps

Given
Radius of the base of the joker’s cap is:\[\small r = 7~\text{cm}\]
Height of the cap is:\[\small h = 24~\text{cm}\]
Find the slant height
Using the relation:\[\small l^2 = r^2 + h^2\]
Substituting the values: \[\small \begin{aligned} l^2 &= 7^2 + 24^2\\ &= 49 + 576\\ &= 625\\ \Rightarrow l &= \sqrt{625}\\ l &= 25~\text{cm} \end{aligned} \]
Find the curved surface area of one cap
Formula for curved surface area:\[\small \text{CSA} = \pi r l\]
Substituting the values: \[\small \begin{aligned} \text{CSA}&=\frac{22}{7} \times 7 \times 25\\ &= 22 \times 25\\ &= 550 \end{aligned} \]
Therefore,\[\small \text{CSA of one cap} = 550~\text{cm}^{2}\]
Find the sheet required for 10 caps
Number of caps: \(\small 10\)
Therefore,\[\small \begin{aligned}\text{Total Area} &= 550 \times 10\\&=5500\end{aligned}\]
Hence, the area of sheet required to make 10 caps is:\[\small 5500~\text{cm}^{2}\]

💡 Answer Final Answer ›

Answer: \(\small \text{The area of sheet required to make 10 caps is: }5500~\text{cm}^{2}\)

🎯 Exam Significance Exam Significance ›

This question demonstrates practical application of surface area in daily life objects.
It combines geometry and mensuration concepts in a multi-step problem.
Such problems are frequently asked in school and board examinations.
Competitive entrance exams often include questions involving cones and coverings.
It strengthens understanding of the Pythagoras theorem and curved surface area formulas.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Slant height of a cone is calculated using: \[\small l^2 = r^2 + h^2 \]
Curved surface area of a cone is: \[\small \pi r l \]
Open conical objects require only curved surface covering.
Total material required for multiple objects is obtained by multiplication.
Final answers for area must always be written in square units.

← Q6

7 / 8 · 88%

Q8 →

NUMERIC3 marks

A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is \(\small \text{₹ }12 \text{ per } \text{m}^2\), what will be the cost of painting all these cones? (Use \(\small \pi = 3.14\) and take \(\small \sqrt{1.04}=1.02\))

📘 Concept & Theory Concept Used ›

Since the cones are hollow, only the outer curved surface is painted. Therefore, we calculate the curved surface area of each cone.

Slant height of the cone is found using the Pythagoras theorem:

\[\small l^2 = r^2 + h^2 \]

Curved surface area of a cone is:

\[\small \text{CSA} = \pi r l \]

Total painting cost is calculated by:

\[\small \text{Cost} = \text{Area} \times \text{Rate} \]

🗺️ Solution Roadmap Step-by-step Plan ›

Convert diameter into radius and centimetres into metres.
Calculate the slant height using the Pythagoras theorem.
Find the curved surface area of one cone.
Calculate painting cost for one cone.
Multiply by 50 to get total painting cost.

📊 Graph / Figure Graph / Figure ›

Fig. 1 — Free body diagram

✏️ Solution Complete Solution ›

Step-by-step Solution · 14 steps

Given
Diameter of the base of each cone is:\[\small 40~\text{cm}\]
Radius is half of the diameter.\[\small r = \frac{40}{2}=20\text{ cm}\]
Converting into metres:\[\small r = \frac{20}{100}~\text{m}=0.2~\text{m}\]
Height of each cone is:\[\small h = 1~\text{m}\]
Find the slant height
Using the relation:\[\small l^2 = r^2 + h^2\]
Substituting the values: \[\small \begin{aligned} l^2 &= (0.2)^2 + (1)^2\\ &= 0.04 + 1\\ &= 1.04\\ \Rightarrow \sqrt{1.04} &= 1.02 \end{aligned} \]
Therefore,\[\small l = 1.02~\text{m}\]
Find the curved surface area of one cone
Formula for curved surface area:\[\small \text{CSA} = \pi r l\]
Using \[\small \pi = 3.14 \]
\[\small \begin{aligned} \text{CSA} &= 3.14 \times 0.2 \times 1.02\\ &=3.14\times 0.2 \times 1.02\\ &=3.14 \times 0.204\\ &= 0.64056~\text{m}^{2}\\ &\approx 0.64~\text{m}^{2} \end{aligned} \]
Find the painting cost for one cone
Cost of painting \(\small 1~\text{m}^{2}\) is ₹12.
Therefore, \[\small \begin{aligned} \text{Cost for one cone} &= 0.64 \times 12\\ &= \text{₹}7.68 \end{aligned} \]
Find the cost for 50 cones
Number of cones:=50
Therefore, \[\small \begin{aligned}\text{Total Cost} &= 7.68 \times 50\\&= 384 \end{aligned}\]

💡 Answer Final Answer ›

Answer: Total cost of painting all the cones is:₹\(\small 384\)

🎯 Exam Significance Exam Significance ›

This question connects mensuration with practical public utility applications.
Students learn importance of unit conversion in geometry problems.
Board examinations frequently include questions involving cones and painting costs.
Competitive exams test accuracy in decimal calculations and area formulas.
The problem strengthens understanding of curved surface area and commercial mathematics.

🔑 Key Takeaways Key Takeaways ›

Key Takeaways · 5 points

Radius is half of the diameter.
Unit conversion from centimetres to metres is essential before calculations.
Slant height of a cone is calculated using: \[\small l^2 = r^2 + h^2 \]
Curved surface area of a cone is: \[\small \pi r l \]
Painting cost problems combine geometry with real-life commercial calculations.

← Q7

8 / 8 · 100%

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