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9.4.3. Cylinders
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Today, we're going to explore cylinders! A cylinder is a 3D shape with two parallel circular bases connected by a curved surface at a specific height. Can anyone tell me where we see cylinders in our everyday lives?
A can of soda!
A water pipe!
Great examples! All these objects are right circular cylinders. Now, let's talk about how to describe the dimensions of a cylinder. We have the radius of the base and the height. Does anyone know what we use to measure the lateral surface area of a cylinder?
I think it involves the radius and the height?
Exactly! To find the lateral surface area, we use the formula LSA = 2πrh. Now, let's remember this with the acronym 'LHR' for Lateral, Height and Radius!
LHR! That makes it easier to remember!
Yes! Now let's summarize: cylinders have circular bases, and we use the radius and height for calculations.
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Now that we have understood how to find the lateral surface area, let’s calculate the total surface area of a cylinder. Who can share the formula?
Is it TSA = 2πr² + 2πrh?
Correct! This combines the areas of the two circular bases and the lateral surface area. Together, the total surface area formula can be simplified to TSA = 2πr(r + h). Can anyone see how this formula relates to our previous formula?
It uses both the radius and height just like the LSA!
Exactly! So, let's remember the mnemonic 'T-R-H' for Total, Radius, and Height!
T-R-H! I’ll remember that!
Now, let's summarize: the total surface area encompasses both circular bases and the curved surface area.
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Create a free accountNow, let’s apply what we’ve learned in real-world contexts. For instance, if we had a cylindrical water tank, how would we determine the area needed to cover it with a painted surface?
We would use the total surface area formula!
Correct! And remember, the practical applications are essential for understanding the value of these calculations. Now let’s tackle a challenge: If our tank has a radius of 3m and a height of 5m, what’s the total area we need to paint?
I think it's TSA = 2πr(r + h), so we plug in the values...
Absolutely! What do you get?
With π ~ 3.14, it’s about 50.24 square meters!
Excellent work! So, summarizing again: for practical applications, we utilize our formulas to find area in real settings.
Overview
Short Summary
This section focuses on the surface area of cylinders, exploring definitions, mathematical formulations, and practical applications.
Medium Summary
The section discusses right circular cylinders in detail, including how to compute their lateral and total surface areas. Practical activities and examples illustrate the concepts to enhance understanding and application in real-world scenarios.
Detailed Summary
Detailed Summary
In this section, we delve into the characteristics and formulas related to the surface area of cylinders, particularly right circular cylinders, which are commonly seen in everyday objects like cans and pipes. We start by outlining the method to determine the lateral surface area and the total surface area of a cylinder.
The lateral surface area can be determined by examining a rectangular strip that, when wrapped around the cylinder, corresponds to its height and the circumference of the base.
The detailed formulas include:
- Lateral Surface Area (LSA) = 2πrh
- Total Surface Area (TSA) = 2πr(r + h)
Furthermore, practical examples help illustrate how to compute the surface areas in real-life contexts, aiding the comprehension of these geometric concepts. Activities are provided to encourage hands-on learning and critical thinking.
Reference YouTube Videos
Audio Book
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Create a free accountMost of the cylinders we observe are right circular cylinders. For example, a tin, round pillars, tube lights, water pipes etc.
Detailed Explanation
A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. Specifically, a right circular cylinder has bases that are directly above one another and are equal in size. Typical examples include everyday objects like soup cans and columns.
Examples & Analogies
Think of a water bottle. The bottle is a right circular cylinder—the top and bottom are circular and flat, and the sides curve around. This shape helps it hold liquid efficiently and makes it easy to grip.
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(i) Take a cylindrical can or box and trace the base of the can on graph paper and cut it. Take another graph paper in such a way that its width is equal to the height of the can. Wrap the strip around the can such that it just fits around the can (remove the excess paper). Tape the pieces together to form a cylinder.
What is the shape of the paper that goes around the can?
Detailed Explanation
In the activity, you take a cylindrical can and trace its circular base on paper. This gives you the dimensions of the base. When you wrap another piece of paper around the can, that piece of paper forms a rectangle—the height of the can becomes the height of the rectangle, and the length of the rectangle equals the circumference of the base circle, which is calculated using the formula 2πr.
Examples & Analogies
Imagine wrapping a label around a can of soda. The label fits perfectly around the can, just like the strip of paper does in this activity. This helps you visualize how the surface of the cylinder can be 'unwrapped' into a flat shape.
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Create a free accountThe lateral (or curved) surface area of a cylinder is 2πrh. Note: We take π to be 22/7 unless otherwise stated. The total surface area of a cylinder = πr² + 2πrh + πr² = 2πr² + 2πrh or 2πr (r + h)
Detailed Explanation
The lateral surface area refers to the area of the curved surface of the cylinder, calculated as 2πrh, while the total surface area includes the area of the two circular bases and the lateral surface area. The formula can be summarized as total surface area = 2πr(r + h), where r is the radius and h is the height of the cylinder.
Examples & Analogies
If you were to paint a cylindrical tank, you would need to know both the area of the cylindrical surface and the top and bottom circular areas. Think of it like wrapping a gift: the wrapping paper must cover not just the sides (the lateral area) but also the top and bottom (the circular bases).
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Create a free accountTRY THESE Find total surface area of the following cylinders.
Detailed Explanation
In this section, students are encouraged to practice calculating the surface area of given cylinders using the formulas learned earlier. By substituting the known values of radius and height into the surface area formula, they can find the lateral and total areas, enhancing their understanding of application in various scenarios.
Examples & Analogies
Imagine you are tasked with covering a cylindrical water tank with paint. You would use the formulas to determine how much paint is needed, effectively translating the mathematical concepts into a real-world task that requires precision and application.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Cylinders: 3D shapes with two circular bases.
Lateral Surface Area (LSA): Calculated using the formula 2πrh.
Total Surface Area (TSA): 2πr² + 2πrh can also be expressed as 2πr(r + h).
Radius and Height: Fundamental dimensions needed for cylinder calculations.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
Example 1: Calculate LSA of a cylinder with radius 3 cm and height 5 cm, which gives LSA = 2π(3)(5) = 30π cm².
Example 2: The TSA of a cylinder with a height of 7 cm and a radius of 4 cm can be computed as TSA = 2π(4)(7) + 2π(4)² = 88π cm².
Memory Aids
Interactive tools to help you remember key concepts
Stories
Flash Cards
Glossary
Cylinder
A 3D shape with two parallel circular bases connected by a curved surface.
Lateral Surface Area (LSA)
The surface area of the curved part of a cylinder, calculated as 2πrh.
Total Surface Area (TSA)
The total area of a cylinder's surface, calculated as 2πr² + 2πrh.
Radius (r)
The distance from the center of a circular base to its perimeter.
Height (h)
The perpendicular distance between the two circular bases of a cylinder.