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Cylinder is an important topic in Mathematics. It is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface.

In this post, we will explore the properties of a cylinder and how to calculate its volume and surface area.

Let’s start with the basic definition of a right circular cylinder.

A cylinder is a solid shape that has two parallel circular bases of equal size and shape.

The curved surface that connects the two bases is called the lateral surface.

The axis of the cylinder is a line passing through the center of both bases.

Types Of Cylinders

(i). Solid Cylinder

(ii). Hollow Cylinder

Area Of A Solid Cylinder

Total Surface Area of Right Circular Cylinder = Curved Surface Area + Cicular Base Area + Circular Top Surface Area.

Since the Cicular Base Area And The Circular Top Surface Area of a cylinder are equal

Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Top Surface Area)

Or

Total Surface Area of Right Circular Cylinder = Curved Surface Area + 2(Cicular Base Surface Area)

Therefore the formula to calculate the surface area of a cylinder is expressed as the following:

$$SA = 2\pi r h + 2\pi r^2$$

where SA is the surface area, r is the radius of the base, and h is the height of the cylinder.

Here, Curved Surface Area, $$CSA = 2\pi r h$$ and

Cicular Base Area = Top Surface Area = $$\pi r^2$$

Volume Of A Solid Cylinder

The formula to calculate the volume of a cylinder is given by:

$$V = \pi r^2 h$$

where V is the volume, r is the radius of the base, and h is the height of the cylinder.

Now let’s take an example to understand how to use these formulas. Suppose we have a cylinder with a radius of 4 cm and a height of 10 cm. To calculate its volume, we can use the formula:

$$V = \pi (4)^2 (10) = 160\pi$$

Therefore, the volume of the cylinder is 160π cubic cm.

To calculate its surface area, we can use the formula:

$$SA = 2\pi (4) (10) + 2\pi (4)^2 = 120\pi$$

Therefore, the surface area of the cylinder is 120π square cm.

In conclusion, understanding the properties of a cylinder and how to calculate its volume and surface area is important in CBSE Class 10 Mathematics. By using the formulas mentioned above, you can easily solve problems related to cylinders.

Hollow Cylinder

A hollow cylinder is a three-dimensional object with a circular base and a cylindrical shape. It is also known as a cylindrical shell. The cylinder has two circular faces and a curved surface. The thickness of the cylinder is uniform and it is hollow from inside.

Volume Of A Hollow Cylinder

The volume of a hollow cylinder can be calculated using the formula V = πh(R²-r²), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.

Surface Area Of A Hollow Cylinder

The surface area of a hollow cylinder can be calculated using the formula A = 2πh(R+r), where h is the height of the cylinder, R is the radius of the outer circle, and r is the radius of the inner circle.

Hollow cylinders are used in various applications such as pipes, drums, and containers. They are also used in engineering structures such as bridges and towers.

In conclusion, a hollow cylinder is a useful shape in various fields and can be easily calculated using mathematical equations.

Cone

A cone is a three-dimensional geometric shape that has a circular base and a single vertex. It can be visualized as a pyramid with a circular base. In this note, we will cover the basic concepts and formulas related to cones.

Surface Area of a Cone

The surface area of a cone is the sum of the areas of its base and lateral surface. The formula to calculate the surface area of a cone is:

$$A = \pi r (r + l)$$

Where:

• ( A ) is the surface area of the cone
• $$\pi = 3.14159$$
• ( r ) is the radius of the base of the cone
• ( l ) is the slant height of the cone

Volume of a Cone

The volume of a cone is the amount of space enclosed by it. The formula to calculate the volume of a cone is:

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

Where:

• ( V ) is the volume of the cone
• ( \pi ) is a mathematical constant approximately equal to 3.14159
• ( r ) is the radius of the base of the cone
• ( h ) is the height of the cone

Example Equations

Here are a few example equations related to cones:

1. Equation for calculating the slant height of a cone: $$l = \sqrt{r^2 + h^2}$$
2. Equation for calculating the radius of a cone given its slant height and height: $$r = \sqrt{l^2 – h^2}$$
3. Equation for calculating the height of a cone given its volume and radius: $$h = \frac{3V}{\pi r^2}$$

Sphere

Solid Sphere

Introduction – A solid sphere is a three-dimensional geometric figure in which all points inside the sphere are at the same distance from its center. – It is a type of 3D shape known as a “sphere” with a uniform density throughout.

Characteristics – The solid sphere has a well-defined volume, surface area, and mass. – It is completely filled with matter.

Volume of Solid Sphere The formula to calculate the volume (\(V_s\)) of a solid sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \] Where: \(V_s\) = Volume of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere.

Surface Area of Solid Sphere The formula to calculate the surface area (\(A_s\)) of a solid sphere is given by: \[ A_s = 4 \pi r^2 \]

Where: \(A_s\) = Surface area of the solid sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(r\) = Radius of the sphere

Mass of Solid Sphere The mass (\(m_s\)) of a solid sphere can be calculated using the formula: \[ m_s = \text{Density} \times V_s \] Where: \(m_s\) = Mass of the solid sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_s\) = Volume of the solid sphere (calculated using the previous formula)

Hollow Sphere

Introduction – A hollow sphere is also a three-dimensional geometric figure, but unlike a solid sphere, it has an empty space inside. – It consists of an outer shell or surface with a certain thickness and an inner empty region.

Characteristics – The hollow sphere has a well-defined outer radius (\(R\)), inner radius (\(r\)), volume, surface area, and mass. – It is partially filled with matter, mainly in the form of the outer shell.

Volume of Hollow Sphere The formula to calculate the volume (\(V_h\)) of a hollow sphere is given by: \[ V_h = \frac{4}{3} \pi (R^3 – r^3) \] Where: – \(V_h\) = Volume of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere

Surface Area of Hollow Sphere The formula to calculate the surface area (\(A_h\)) of a hollow sphere is given by: \[ A_h = 4 \pi (R^2 – r^2) \] Where: \(A_h\) = Surface area of the hollow sphere, \(\pi\) (\(\pi\)) ≈ 3.14159, \(R\) = Outer radius of the sphere, \(r\) = Inner radius of the sphere

Mass of Hollow Sphere The mass (\(m_h\)) of a hollow sphere can be calculated using the formula: \[ m_h = \text{Density} \times V_h \] Where: \(m_h\) = Mass of the hollow sphere, \(\text{Density}\) = Density of the material making up the sphere (usually in \(kg/m^3\)), \(V_h\) = Volume of the hollow sphere (calculated using the previous formula)