Chapter 11 · CBSE · Class IX
Table of Contents
13 sections
- 1 Introduction ›
- 2 Definition ›
- 3 Important Terms Related to Cone ›
- 4 Relation Between Radius, Height and Slant Height ›
- 5 Curved Surface Area of a Cone ›
- 6 Derivation of Curved Surface Area Formula ›
- 7 Total Surface Area of a Cone ›
- 8 Important Formula Chart ›
- 9 Solved Examples ›
- 10 CBSE Case Study Based Question ›
- 11 Exam Tip ›
- 12 Common Mistakes Students Make ›
- 13 Quick Revision ›
📖 Introduction
📘 Definition
🎨 SVG Diagram
Labeled Diagram of a Right Circular Cone
📌 Note
Important Terms Related to Cone
🗒️ Relation Between Radius, Height and Slant Height
Relation Between Radius, Height and Slant Height
In a right circular cone, radius, height and slant height form a right-angled triangle. Therefore, by Pythagoras Theorem:
\[\small l^2=h^2+r^2 \]Hence,
\[\small \boxed{l=\sqrt{h^2+r^2}} \]This formula is extremely important because surface area formulas of a cone depend on slant height.
In board examinations, students often forget to calculate slant height before applying
the CSA or TSA formula.
📘 Definition
Curved Surface Area of a Cone
📐 Derivation
Derivation of Curved Surface Area Formula
When the curved surface of a cone is opened, it forms a sector of a circle of radius \(\small l\).
The arc length of the sector equals the circumference of the cone’s base.
\[\small \text{Arc length}=2\pi r \]Radius of the sector:
\[\small =l \]Area of the complete circle of radius \(\small l\):
\[\small \pi l^2 \]Therefore,
\[\small \frac{\text{Area of sector}}{\pi l^2} = \frac{2\pi r}{2\pi l} \] \[\small \frac{\text{Area of sector}}{\pi l^2} = \frac{r}{l} \] \[\small \text{Area of sector} = \pi l^2 \times \frac{r}{l} \] \[\small \boxed{\text{CSA}=\pi rl} \]
🔢 Formula
Total Surface Area of a Cone
🔢 Formula
Important Formula Chart
✏️ Example
Solved Examples
Find the curved surface area of a cone whose radius is \(\small 7\text{ cm}\) and
slant height is \(\small 10\text{ cm}\).
Curved Surface Area of Cone:
\[\small \text{CSA}=\pi rl \]- Write the given values.
- Apply the CSA formula.
- Simplify carefully.
-
given
\[\small r=7\text{ cm}, \quad l=10\text{ cm}\]
-
\(\small \text{CSA}=\pi rl\)
\[\small \begin{aligned}&=\frac{22}{7}\times 7\times 10\\ &=220\text{ cm}^2\end{aligned}\]
-
Therefore,
\[\small \boxed{\text{CSA}=220\text{ cm}^2}\]
A cone has radius \(\small 6\text{ cm}\) and height \(\small 8\text{ cm}\).
Find its total surface area.
First calculate slant height using Pythagoras theorem.
-
Slant Height
\[\small \begin{aligned} l&=\sqrt{h^2+r^2}\\ &=\sqrt{8^2+6^2}\\ &=\sqrt{64+36}\\ &=\sqrt{100}\\ l&=10\text{ cm} \end{aligned} \]
-
Total Surface Area:
\[\small \begin{aligned} \text{TSA}&=\pi r(l+r)\\ &=\frac{22}{7}\times 6\times (10+6)\\ &=\frac{22}{7}\times 96\\ &=301.71\text{ cm}^2 \end{aligned} \]
-
Therefore,
\[\small \boxed{\text{TSA}\approx 301.71\text{ cm}^2}\]
📋 Case Study
CBSE Case Study Based Question
A company manufactures party caps in the shape of cones. Each cap has radius \(\small 5\text{ cm}\) and slant height \(\small 12\text{ cm}\). The company wants to know the amount of colored paper required to make one cap.
Think and Analyze
Since the cap is open at the bottom, only the curved surface area is needed.
Solution
\[\small \text{CSA}=\pi rl \] \[\small =\frac{22}{7}\times 5\times 12 \] \[\small =188.57\text{ cm}^2 \]Therefore, the paper required is:
\[\small \boxed{188.57\text{ cm}^2} \]
⚡ Exam Tip
❌ Common Mistakes
Common Mistakes Students Make
- Using height instead of slant height in CSA formula.
- Forgetting to add base area while calculating TSA.
- Writing incorrect square units.
- Incorrect use of Pythagoras theorem.
- Rounding answers too early in calculations.
⚡ Quick Revision
- A cone has one circular base and one curved surface.
- Slant height: \[\small l=\sqrt{h^2+r^2} \]
- Curved Surface Area: \[\small \pi rl \]
- Total Surface Area: \[\small \pi r(l+r) \]
- Surface area is measured in square units.