Class 9 Maths – Surface Areas and Volumes

11.1 Surface Area of a Right Circular Cone

• Cone: Solid with a circular base and a vertex not in the plane of the base.
• Elements:
  • Radius of base = r
  • Slant height = l
  • Height = h

Formulas:

1. Curved Surface Area (CSA):

$$CSA = \pi r l$$CSA=πrl

2. Total Surface Area (TSA):

$$TSA = \pi r (l + r)$$TSA=πr(l+r)

Note: $l = \sqrt{r^2 + h^2}$l=r2+h2​

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11.2 Surface Area of a Sphere

• Sphere: Set of points in space equidistant from a centre.
• Radius: r

Formulas:

1. Surface Area (SA) of sphere:

$$SA = 4 \pi r^2$$SA=4πr2

2. Curved Surface Area of hemisphere:

$$CSA = 2 \pi r^2$$CSA=2πr2

3. Total Surface Area of hemisphere:

$$TSA = 3 \pi r^2$$TSA=3πr2

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11.3 Volume of a Right Circular Cone

• Formula:

$$V = \frac{1}{3} \pi r^2 h$$V=31​πr2h

• Explanation: Volume = 1/3 × Base Area × Height

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11.4 Volume of a Sphere

• Formula:

$$V = \frac{4}{3} \pi r^3$$V=34​πr3

• Volume of hemisphere:

$$V = \frac{2}{3} \pi r^3$$V=32​πr3

Notes:

• π ≈ 3.1416 or use 22/7
• Units of surface area = square units
• Units of volume = cubic units

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Quick Short Q&A (Most Possible)

Question	Short Answer
CSA of cone formula?	π r l
TSA of cone formula?	π r (l + r)
Slant height formula?	l = √(r² + h²)
SA of sphere?	4 π r²
TSA of hemisphere?	3 π r²
CSA of hemisphere?	2 π r²
Volume of cone?	(1/3) π r² h
Volume of sphere?	(4/3) π r³
Volume of hemisphere?	(2/3) π r³
Unit of volume?	Cubic units