11. Surface Areas and Volumes

Theorem 11.1

The curved surface area of a cone is given by \( \pi r l \), where \( r \) is the base radius and \( l \) is the slant height.

Theorem 11.2

The total surface area of a cone, including its base, is \( \pi r (l + r) \).

Exercise 11.1

1. Find the total surface area of a cone if its slant height is 21 m and base diameter is 24 m.
2. A conical tent is 10 m high with a base radius of 24 m. Find its slant height.
3. Calculate the cost of canvas required to make the tent if the cost per square meter is ₹70.

11.2 Surface Area of a Sphere

Theorem 11.3

The surface area of a sphere is given by \( 4\pi r^2 \), where \( r \) is the radius.

Exercise 11.2

1. Find the surface area of a sphere with diameter 14 cm.
2. A hemispherical dome needs to be painted. If the cost of painting is ₹20 per square meter, calculate the total cost.

11.3 Volume of a Right Circular Cone

Theorem 11.4

The volume of a cone is given by \( \frac{1}{3} \pi r^2 h \), where \( r \) is the base radius and \( h \) is the height.

11.4 Volume of a Sphere

Theorem 11.5

The volume of a sphere is given by \( \frac{4}{3} \pi r^3 \), where \( r \) is the radius.

Exercise 11.3

1. Find the volume of a sphere with radius 10 cm.
2. A hemispherical tank has an inner radius of 1 m and a thickness of 0.25 cm. Calculate the volume of iron used to make the tank.

11.5 Summary