What is a Cylinder
In mathematics, a cylinder is a three-dimensional solid object that consists of two congruent circular bases and a curved lateral surface that connects them. The bases are parallel, and the lateral surface is perpendicular to the bases.
It can be considered a stack of circles, with each circle on top of the one below it. All the circles share a central axis. The Radius of the Cylinder is the measurement around each base taken from its center, and the height is the separation between the two bases measured perpendicularly.
Define Cylinder
“A cylinder is a 3D geometric shape. It has two congruent, parallel circular bases and a curved lateral surface connecting them. The volume and surface area can be calculated using mathematical formulas.”
Graduated cylinder
It is a tall, narrow-shaped laboratory glassware used to measure and dispense liquids accurately. It has a slim cylindrical shape with a spout for easy pouring and a series of calibrated marks etched along the side to indicate the volume of liquid contained within it. It is designed to be read at eye level to ensure accurate measurement. It is commonly used in chemistry, biology, and other scientific fields where precise liquid volume measurements are required.
Volume of Cylinder
The amount of space a cylinder has enclosed within it is its volume. It is computed by dividing the height by the area of its base, which is typically round. The following is the formula for a volume:
V = πr²h
where;
V = volume
h = height
r = radius
Example
Find the Volume of the cylinder if the radius is 4 units & height is 8 units.
Solution:
According to the formula of volume V = πr²h
V = π(4)²(8)
V = π(16)(8)
V = 128π cubic units
Therefore, the volume is 128π cubic units.
Lateral Surface Area
A cylinder’s lateral surface area is the portion of its curved surface, excluding the portions that make up the top and bottom circular bases. The following formula calculates the lateral surface area:
L = 2πrh
where;
h = height
r = radius
Example
Find the lateral surface area of the cylinder if the radius is 4 units & height is 8 units.
Solution:
According to the formula of lateral surface area L = 2πrh
L = 2π(4)(8)
L = 64π square units
Therefore, the lateral surface area is 64π square units.
Total Surface area
The combined surface areas of the top and bottom circular bases, as well as the curved surface (lateral surface), make up a cylinder. The following equation determines total surface area:
A = 2πr(r+h)
where;
h = height
r = radius
Example
Find the Total surface area of the cylinder if the radius is 4 units & height is 8 units.
Solution:
According to the formula of Total surface area A = 2πr(r+h)
A = 2π(4)(4+8)
A = 2π(4)(12)
A = 96π square units
Therefore, the total surface area is 96π square units.
Shapes of Cylinder
A cylinder’s shape is defined by its two circular bases and the curved lateral surface that connects them. The two circular bases are congruent and parallel, and the lateral surface is curved and perpendicular to the bases. We can see the lateral surface as a rectangle wrapped around it, with its length equal to the base’s circumference and its height equivalent to its height. No of their size or position, they all have the same shape.
Cross-sectional shapes of Cylinder
The cross-sectional shape of a cylinder is its shape when cut perpendicular to its central axis. Its form, which might be circular or non-circular, dictates the cylinder’s profile. Here are some common cross-sectional conditions and their names:
- Right circular
- Oblique
- Elliptical
- Parabolic
- Hyperbolic
Right circular cylinder
It is called a right circular cylinder if its base is a circle. Its axis is perpendicular to the base. Since the cylinder is the limiting form of a prism, the volume and surface area is calculated by the same prism formula.
Volume of a Right circular cylinder
Volume = Area of base x Height
Volume = π r2 h = (πd/4) x h
Lateral Surface Area
Lateral Surface Area = Perimeter of the base x height
Lateral Surface Area = 2πrh
Lateral Surface Area = πdh
Total Surface area
Total Surface Area = Lateral Surface Area + Area of bases
Total Surface Area = 2πrh + 2πr2
Total Surface Area = 2πr (h+r)
Example
The radius of a right circular cylinder is 25 cm, and its height is 15cm. Find its volume, lateral surface, and the whole surface area.
Solution:
Here r = 25cm, h = 15cm
Volume = π r2 h
Volume = π x 625 x 15
Volume = 29452.43 cu. m.
Lateral Surface Area = 2πrh
Lateral Surface Area = π x 40 x 15
Lateral Surface Area = 2356.19 sq. sm.
Total surface area =Lateral surface area + Area of bases
Total Surface Area = 2πr (h+r)
Total Surface Area = π x 50(25+15)
Total Surface Area = 6238.185 sq. cm
Oblique cylinder
It is one in which the axis is not perpendicular to the bases. In other terms, instead of standing vertically oriented, as in the case of a conventional upright cylinder, it is orientated at an angle or slant. The bases are still parallel and congruent circles, but they are not perpendicular to the sides of them. They are less standard than vertical cylinders, but they exist in specific applications, such as engineering, architecture, and design. In contrast, it has a straight and perpendicular axis, making it easier to visualize and work mathematically.
Hollow Circular Cylinder
A three-dimensional object called a hollow circular cylinder has the appearance of a cylinder with round cross-sections but is hollow on the interior. It is created by deducting the volume of a smaller from that of a bigger cylinder along the same axis. The resulting shape features a cylindrical wall that is continuously spaced from the center and a hollow core. The difference between the two radii determines the wall’s thickness. They are often used in engineering, construction, and manufacturing to create pipes, tubes, and other hollow cylindrical constructions with a specific thickness or diameter.
Volume of a Hollow circular cylinder
Volume = Volume of external – Volume of internal cylinder
Volume = π R2 h – π r2 h
Volume = π (R2 – r2 ) h
Lateral Surface Area
Lateral Surface Area = External surface area + Internal surface area
Lateral Surface Area = 2πRh + 2πrh
Lateral Surface Area = 2π (R +r) h
Total Surface Area
Total Surface Area = Lateral Area + Area of solid bases
Total Surface Area = 2π (R +r) h + 2π (R2 +r2)
Example
Find the weight, lateral surface area, and total surface area of iron pipe whose interior and exterior diameters measure 15cm and 17cm, respectively, and length 10m; one cubic cm of iron weighing 0.8gm.
Solution:
d=15cm
D = 17cm
h=10cm= 1000cm
r = 7.5cm
R=8.5cm
Volume = π (R2 – r2 ) h
Volume = π x (72.25-64.75)1000
Volume = 2346.19 cu. cm.
Weight = Volume x density = 23561.9 x 0.8
Weight =18849.52gms
Lateral Surface Area = 2π (R +r) h
Lateral Surface Area =2π(R+r)h
Lateral Surface Area =2π(8.5+ 7.5)1000
Lateral Surface Area =2π x 16 x 1000
Lateral Surface Area =100530.96 sq. cm.
Total Surface Area = 2π (R +r) h + 2π (R2 +r2)
Total Surface Area = 100530.96+47.12
Total Surface Area = 100578.08 sq. cm.
Elliptical cylinder
It is a three-dimensional geometry having elliptical cross-sections. It resembles a round cylinder, but the cross-sections are now elliptical rather than circular. It has two parallel, congruent elliptical bases and a curved lateral surface connecting them. Their curved lateral surface is not straight, and it bulges out more in some places than in others.
Volume of Elliptical cylinder
Volume = Area of the base x height
Volume = πabh
Lateral Surface Area
Lateral Surface Area = perimeter of base x height
Lateral Surface Area = π (a+b) h
Total Surface Area
Total Surface Area = Lateral Surface Area + Area of bases
Total Surface Area = π (a+b) h + 2πab
Example
Parabolic cylinder
A three-dimensional shape is known a parabolic cylinder having a parabolic cross-section. It is formed by extruding a parabola along a straight line to create a curved surface. The resulting shape is a cylinder with a parabolic cross-section that bulges out more in the middle and is flatter at the ends. They are used in various applications, such as in the design of reflectors and antennas, where the parabolic shape can focus light or radio waves to a point.
Hyperbolic cylinder
It is a three-dimensional shape having hyperbolic cross-sections. It is formed by extruding a hyperbola along a straight line to create a curved surface. The resulting shape is a cylinder with two parallel, congruent hyperbolic bases and a curved lateral surface connecting them. Their lateral surface has a saddle-like form and curves in two separate directions. They are used in various applications, such as in architecture, where they can be used to create unique and exciting structures.
Frustum of a Right Circular Cylinder
When a plane parallel cuts a right circular cylinder to its base (or perpendicular to its axis), the section is called a cross-section, which is a circle. If the plane section is not parallel to the bases, i.e., it is oblique, the portion between the plane section and the base is called the Frustum of the right circular cylinder. This cutting section is an ellipse.
The given figure represents a frustum of the cylinder whose cutting plane is inclined at an angle θ to the horizontal.
Volume of Frustum of a Right Circular Cylinder
If r is the radius of the base and ha is the average height of the Frustums, then:
Volume = Area of base x average height
Volume = πr²ha
Lateral surface area
Lateral surface area = Perimeter of the base x average height
Lateral surface area = 2π rha
Total surface area
Total surface area = Area of the base + Area of the ellipse +Lateral surface area
For the ellipse, AB=AC Cos θ
AB = (AB/ Cos θ) = (2r/Cos θ)
So, the semi-mirror axis = r
and the semi-major axis = (AB/2) = (r/Cos θ)
Hence the area of the ellipse = π a b= πr²/ Cos θ
Types of cylinder
There is just one sort of cylinder in geometry, a three-dimensional figure of two parallel, congruent circular bases and a curving lateral surface bridging them. Nonetheless, They can be categorized into practical applications depending on their characteristics, intended use, and other elements. In daily life, following popular types are frequently used:
- Hydraulic
- Pneumatic
- Internal combustion engine
- Welded
- Telescopic
- Rotary
- Double-acting
Hydraulic cylinder
A hydraulic cylinder is a mechanical actuator providing a unidirectional force through a stroke. It is also sometimes called a hydraulic ram or hydraulic actuator. It consists of a piston and a cylinder barrel, both made of metal. The piston moves back and forth within the cylinder barrel, driven by a hydraulic fluid pressurized by a hydraulic pump.
They are frequently employed in various industrial applications, such as building, manufacturing, and agriculture machinery. They often provide linear motion to other components, such as a boom arm or a plow blade. They are perfect for applications requiring a lot of energy since they can produce a lot of force.
They come in various designs, including single-acting and double-acting cylinders. Single-acting cylinders use hydraulic pressure to extend the piston but rely on a mechanical spring or other external force to retract it. Double-acting cylinders can extend and retract the piston using hydraulic pressure, allowing for more precise control over the motion of the cylinder.
Pneumatic cylinder
A pneumatic cylinder is a mechanical device that uses compressed air to produce force and motion linearly. It is sometimes referred to as an air cylinder or pneumatic actuator. It consists of a piston and a cylinder barrel, both made of metal. The piston moves back and forth within the cylinder barrel, driven by compressed air controlled by a pneumatic valve.
They are commonly used in various applications, including automation, manufacturing machinery, and transportation equipment. They often provide linear motion to other components, such as a conveyor belt or a robotic arm.
Single-acting cylinders use compressed air to extend the piston but rely on a mechanical spring or other external force to retract it. Double-acting cylinders can extend and retract the piston using compressed air, allowing for more precise control over its motion of it. Because they are frequently more straightforward and less expensive than hydraulic cylinders, they are preferred for many industrial applications.
Internal combustion engine cylinder
A piston that oscillates back and forth in an internal combustion engine cylinder transforms the chemical energy of fuel into mechanical energy. Its wall is typically made of cast iron and coated with a layer of a unique material to reduce friction.
The high-pressure gas produced when fuel is ignited in it forces the piston downward, rotating the crankshaft and having power. The energy and air combination is then compressed as the piston rises back to the top of it, where it is ignited once more to complete the cycle.
Most internal combustion engines have multiple cylinders arranged in a row, a V-shape, or a flat configuration. The number of cylinders and their arrangement affects the engine’s power, efficiency, and smoothness of operation.
The displacement or size of the cylinder plays a crucial role in influencing the engine’s production of power. The amount of space the piston passes through throughout its stroke is known as displacement. Engine displacement is typically measured in liters or cubic inches and is used to classify engines as small or large.
Welded cylinder
A welded cylinder is a type of hydraulic cylinder created by welding two end caps onto a cylinder barrel to create a sealed, pressurized chamber. The barrel and end caps are often welded together using a high-temperature welding procedure and are typically composed of steel or other robust materials.
Welded cylinders are commonly used in heavy-duty industrial applications, such as construction equipment, mining machinery, and material handling systems. They are ideal for applications requiring high pressure and force, as they can withstand extreme conditions and provide reliable performance.
The sizes and layouts of the range can be single-acting or double-acting. Single-acting welded cylinders are pressurized on one side of the piston and rely on an external force, such as gravity or spring, to return the piston to its original position. Double-acting welded cylinders are pressurized on both sides of the piston, allowing for precise control of the piston’s motion.
Industrial applications like welded cylinders because they are reliable, simple to maintain, and adaptable enough to satisfy unique application needs. They are appropriate for hostile situations because they are made to endure high pressures and intense temperatures.
Telescopic cylinder
A hydraulic or pneumatic actuator that consists of a succession of nested cylinders with progressively smaller diameters is called a telescopic cylinder, also known as a multistage or a sleeved cylinder. These nested cylinders, also known as stages, are connected by a series of rods, which extend or retract as it is actuated.
They are used in various applications, such as dump trucks, cranes, and hydraulic presses, where a long stroke length is required in a relatively compact space. The nested design of it allows it to achieve a longer stroke length than a traditional single stage of the same overall length.
The number of stages in it can vary depending on the application but typically ranges from two to six steps. The maximum stroke length achievable with it is limited by the stability of the cylinder and the available space for the nested stages to retract when fully extended.
Rotary cylinder
A rotary cylinder, also called a rotary actuator, transforms fluid power into rotating motion. It is commonly used in automation and robotics to control the position and movement of various components.
They typically consist of a cylindrical housing with an internal piston that rotates as fluid pressure is applied to it. The shaft that extends from one end of it and delivers rotational motion to the controlled components is attached to the piston.
They are available, including rack and pinion, vane-type, and spiral-type. Rack and pinion cylinders convert linear and rotational motions using a gear system. In contrast, vane-type use vanes that slide in and out of slots in the cylinder to produce rotation. Helical-type uses a screw-like mechanism to convert linear motion into rotation.
The specific type of rotary cylinder used depends on application requirements, such as torque, speed, and precision. Rotary cylinders are commonly used in indexing, rotary clamping, and material handling applications.
Double-acting cylinder
A hydraulic or pneumatic cylinder with two ports, one for expansion and the other for contraction, is known as a double-acting cylinder. They can produce force in both directions and can be used in applications requiring precise motion control.
The piston inside it is connected to a rod that can extend or retract, producing a linear motion. A rod that extends or retracts from the cylinder is attached to the piston within, creating a linear motion.
They are commonly used in applications requiring force in both directions, such as hydraulic presses, construction equipment, and material handling. They are also used in applications where precise motion control is needed, such as robotics and automation.
One advantage of using a double-acting cylinder is that it can produce a constant force in both directions. This is so that the forces acting on it are balanced by the pressure applied to both piston sides. They are versatile and commonly used components in various industries because they may also be used to adjust the speed and position of the piston.
FAQ’s
How Hydraulic cylinder repair?
A hydraulic cylinder must be disassembled to be repaired, and each component must be checked for wear or damage. Then, replace any damaged parts, such as seals, rods, or piston heads. Reassemble the cylinder afterward, then test it to ensure it works properly. Finally, re-install the repaired cylinder back into the hydraulic system.
What is the difference between area and volume?
A 2-dimensional surface or region’s area is measured, often in square measurements like square meters or square feet. The space occupied by a three-dimensional item is quantified by its volume, commonly expressed in cubic units like cubic meters or cubic feet. The measurement of space enclosed by a solid object is called volume, area is a measure of the size of a flat surface.
Does a cylinder have 2 or 3 faces?
A cylinder has three faces: two circular faces and one curved rectangular face that connects the two circular faces. The circular faces are parallel and congruent, while the curved rectangular face is perpendicular to the circular faces. The two circular faces are also called the bases of the cylinder.
What is base area?
A three-dimensional object’s base area is its flat bottom surface area. It is calculated by finding the product of the length and width of the object’s base. The base area is essential in calculating the volume and other properties of the object.
What is the flat surface of a cylinder?
The flat surface refers to one of the circular faces that serve as the ends of the cylinder. Two of these circular faces, which are parallel and congruent, make up a cylinder. They are joined by a curved rectangular face, which creates the cylinder’s curved portion. The bases are another name for the flat surfaces of the object.
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