csa of cone class 9 .pptx

ppt on csa of cone

1. UNDERSTANDING AND CALCULATING Surface AREA of cone

2. INTRODUCTION DEFINITION OF CONE :- A three-dimensional geometric figure with a circular base that tapers smoothly to a point called the apex. IMPORTANCE OF CONE :- Understanding the surface area helps in various real-world applications, such as material estimation, manufacturing, and design . OVERVIEW OF THE PRESENTATION :- We'll explore the components of a cone, delve into the surface area formula, discuss its derivation, solve examples, and examine real-world applications.

3. Base :- The circular bottom of the cone. Apex (Vertex):- The point at the top of the cone s . Slant Height :-The ditance from the apex to any point on the perimeter of the base. Height (Altitude): -The vertical distance from the apex to the base. PARTS OF CONE

4. CURVED SURFACE AREA C SA = πrl Where, r :- Radius of the base l :- Slant height

5. GEOMETRIC EXPLANATION OF CONE ℓ 2 π l 2 π r

6. Step-by-Step Derivation If we divide the sector into hundreds of little triangles ,so small that base will be considered a line. Then, Now the area of each triangle = ½ × base of each triangle × l = ½ b₁l + ½ b₂l + ½ b₃l +…. = ½ l (b₁ + b₂ + b₃ + …) = ½ × l × length of entire boundary Curved Surface Area = ½ × l× 2 π r = π rl

7. Example Problem- 1 Q:- A cone with specific dimensions: 10cm 14cm Diameter of the Base: 10 cm Slant height: 14 cm

8. Example Problem- 2 Q:- A cone with specific dimensions: 6 cm 8 cm Radius of the Base: 6 cm Height: 8 cm ( HINT: l = √r² + h²)

9. TOtal SURFACE AREA The total surface area (TSA) of a cone is the sum of its curved surface area (CSA) and the area of its base. The formula for the total surface area ( TS A ) of a cone is TSA = πr ( r + l ), where r is the radius of the base, and l is the slant height .

10. Derivation of TSA= πr (r + l) 1. Curved Surface Area (CSA) Component: The curved surface area ( CSA ) of the cone is given by π r l , as explained earlier . 2. Base Area Component: The base of the cone is a circle, and its area is given by 2 π r 2 . 3. Adding CSA and Base Area: Combining the curved surface area and the base area gives the total surface area (TSA). Mathematical Steps: Start with =CSA= π r l. Add the base area: TSA=CSA+ π r2. Substitute CSA: TSA= π r l+ π r2. Factor out π r: TSA= π r(l + r).

11. Derivation of TSA= πr (r + l)

12. Example Problem- 3 Q:- A cone with specific dimensions: 10 cm 12 cm Diameter of the Base:- 10 cm Height:- 12 cm ( HINT: l = √r² + h²)

13. Example Problem- 4 Q:- A cone with specific dimensions: 6 cm 8cm Radius of the Base: 6 cm Slant height: 8 cm

14. SUMMARY 1. Components of a Cone: Circular base, apex (vertex), slant height, and height. 2. Curved Surface Area Formula: CSA = π r l 3. Total Surface Area Formula: TSA = π r ( r + l )

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