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Math Notes For Class 8 Chapter 5 Ex 5.1 Solved & Quiz
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Math Notes For Class 8 Chapter 5 Ex 5.1 Solved & Quiz
In three-dimensional geometry, a right circular cylinder is a solid figure bounded by a cylindrical surface and two parallel circular bases. Understanding the surface area and volumetric capacity of cylinders is essential for solving real-world mensuration problems in physics, engineering, and architecture.
| Geometric Property | Mathematical Formula | Variables Description |
|---|---|---|
| Curved Surface Area (CSA) | $$\text{CSA} = 2\pi r h$$ | $r = \text{base radius}$, $h = \text{height}$ |
| Area of Two Circular Bases | $$\text{Base Area} = 2\pi r^2$$ | $r = \text{base radius}$ |
| Total Surface Area (TSA) | $$\text{TSA} = 2\pi r(r + h)$$ | $r = \text{base radius}$, $h = \text{height}$ |
| Volume of Solid Cylinder ($V$) | $$V = \pi r^2 h$$ | $r = \text{base radius}$, $h = \text{height}$ |
| Volume of Hollow Cylinder ($V_{\text{hollow}}$) | $$V = \pi (R^2 - r^2) h$$ | $R = \text{outer radius}$, $r = \text{inner radius}$, $h = \text{length/height}$ |
Question 1 (Part i): Find the total surface area and volume of a solid cylinder having base radius $r = 7\text{ cm}$ and height $h = 10\text{ cm}$. Take $\pi = \frac{22}{7}$.
Solution:
Given data:
Radius of the cylinder ($r$) = $7\text{ cm}$
Height of the cylinder ($h$) = $10\text{ cm}$
Step 1: Calculate Total Surface Area (TSA)
The formula for the total surface area of a cylinder is:
$$\text{TSA} = 2\pi r (r + h)$$
Substitute the given values into the formula:
$$\text{TSA} = 2 \times \frac{22}{7} \times 7 \times (7 + 10)$$
$$\text{TSA} = 44 \times 17 = 748\text{ cm}^2$$
Step 2: Calculate Volume ($V$)
The formula for the volume of a cylinder is:
$$V = \pi r^2 h$$
Substitute the given values into the formula:
$$V = \frac{22}{7} \times (7)^2 \times 10$$
$$V = \frac{22}{7} \times 49 \times 10 = 22 \times 7 \times 10 = 1540\text{ cm}^3$$
Answer: Total Surface Area = $$748\text{ cm}^2$$, Volume = $$1540\text{ cm}^3$$
Question 1 (Part ii): Find the total surface area and volume of a solid cylinder having base radius $r = 3.5\text{ cm}$ and height $h = 12\text{ cm}$. Take $\pi = \frac{22}{7}$.
Solution:
Given data:
Radius ($r$) = $3.5\text{ cm} = \frac{7}{2}\text{ cm}$
Height ($h$) = $12\text{ cm}$
Step 1: Calculate Total Surface Area (TSA)
$$\text{TSA} = 2\pi r(r + h)$$
$$\text{TSA} = 2 \times \frac{22}{7} \times \frac{7}{2} \times (3.5 + 12)$$
$$\text{TSA} = 22 \times 15.5 = 341\text{ cm}^2$$
Step 2: Calculate Volume ($V$)
$$V = \pi r^2 h$$
$$V = \frac{22}{7} \times \left(\frac{7}{2}\right)^2 \times 12$$
$$V = \frac{22}{7} \times \frac{49}{4} \times 12 = 22 \times 7 \times 3 = 462\text{ cm}^3$$
Answer: Total Surface Area = $$341\text{ cm}^2$$, Volume = $$462\text{ cm}^3$$
Question 2 (Part i): Find the curved surface area and total surface area of a closed cylindrical tank whose diameter is $14\text{ m}$ and height is $15\text{ m}$. Take $\pi = \frac{22}{7}$.
Solution:
Given data:
Diameter ($d$) = $14\text{ m} \implies \text{Radius } (r) = \frac{d}{2} = \frac{14}{2} = 7\text{ m}$
Height ($h$) = $15\text{ m}$
Step 1: Calculate Curved Surface Area (CSA)
$$\text{CSA} = 2\pi r h$$
$$\text{CSA} = 2 \times \frac{22}{7} \times 7 \times 15 = 44 \times 15 = 660\text{ m}^2$$
Step 2: Calculate Total Surface Area (TSA)
$$\text{TSA} = 2\pi r(r + h)$$
$$\text{TSA} = 2 \times \frac{22}{7} \times 7 \times (7 + 15)$$
$$\text{TSA} = 44 \times 22 = 968\text{ m}^2$$
Answer: Curved Surface Area = $$660\text{ m}^2$$, Total Surface Area = $$968\text{ m}^2$$
Question 2 (Part ii): Find the curved surface area and total surface area of a cylinder with radius $r = 10.5\text{ cm}$ and height $h = 20\text{ cm}$. Take $\pi = \frac{22}{7}$.
Solution:
Given data:
Radius ($r$) = $10.5\text{ cm} = \frac{21}{2}\text{ cm}$
Height ($h$) = $20\text{ cm}$
Step 1: Calculate Curved Surface Area (CSA)
$$\text{CSA} = 2\pi r h$$
$$\text{CSA} = 2 \times \frac{22}{7} \times \frac{21}{2} \times 20 = 22 \times 3 \times 20 = 1320\text{ cm}^2$$
Step 2: Calculate Total Surface Area (TSA)
$$\text{TSA} = 2\pi r(r + h)$$
$$\text{TSA} = 2 \times \frac{22}{7} \times \frac{21}{2} \times (10.5 + 20)$$
$$\text{TSA} = 66 \times 30.5 = 2013\text{ cm}^2$$
Answer: Curved Surface Area = $$1320\text{ cm}^2$$, Total Surface Area = $$2013\text{ cm}^2$$
Question 3: A cylindrical water tank has a base radius of $2.1\text{ m}$ and a height of $5\text{ m}$. Calculate its capacity in liters, knowing that $1\text{ m}^3 = 1000\text{ liters}$.
Solution:
Given data:
Radius ($r$) = $2.1\text{ m} = \frac{21}{10}\text{ m}$
Height ($h$) = $5\text{ m}$
Step 1: Find the Volume of the Tank ($V$)
$$V = \pi r^2 h$$
$$V = \frac{22}{7} \times \left(\frac{21}{10}\right)^2 \times 5$$
$$V = \frac{22}{7} \times \frac{441}{100} \times 5$$
$$V = \frac{22 \times 63 \times 5}{100} = \frac{6930}{100} = 69.3\text{ m}^3$$
Step 2: Convert Volume to Liters
$$\text{Capacity in liters} = 69.3 \times 1000 = 69,300\text{ liters}$$
Answer: Capacity = $$69,300\text{ liters}$$
Question 4: Calculate the height of a solid cylinder whose volume is $1540\text{ cm}^3$ and base radius is $7\text{ cm}$. Take $\pi = \frac{22}{7}$.
Solution:
Given data:
Volume ($V$) = $1540\text{ cm}^3$
Radius ($r$) = $7\text{ cm}$
Height ($h$) = ?
Step 1: Apply Volume Formula
$$V = \pi r^2 h$$
$$1540 = \frac{22}{7} \times (7)^2 \times h$$
$$1540 = \frac{22}{7} \times 49 \times h$$
$$1540 = 154 \times h$$
Step 2: Solve for Height ($h$)
$$h = \frac{1540}{154} = 10\text{ cm}$$
Answer: Height = $$10\text{ cm}$$
Question 5: A cylindrical iron pipe open at both ends has an outer radius of $5\text{ cm}$, an inner radius of $4\text{ cm}$, and a length of $21\text{ cm}$. Find the volume of the iron used in making the pipe.
Solution:
Given data:
Outer radius ($R$) = $5\text{ cm}$
Inner radius ($r$) = $4\text{ cm}$
Length/Height ($h$) = $21\text{ cm}$
Step 1: Apply Hollow Cylinder Volume Formula
$$V = \pi (R^2 - r^2) h$$
$$V = \frac{22}{7} \times (5^2 - 4^2) \times 21$$
$$V = \frac{22}{7} \times (25 - 16) \times 21$$
$$V = 22 \times 9 \times 3$$
$$V = 594\text{ cm}^3$$
Answer: Volume of iron = $$594\text{ cm}^3$$
Question 6: Find the cost of painting the outer curved surface of a cylindrical pillar with base radius $0.7\text{ m}$ and height $4\text{ m}$ at the rate of Rs. 150 per $\text{m}^2$.
Solution:
Given data:
Radius ($r$) = $0.7\text{ m} = \frac{7}{10}\text{ m}$
Height ($h$) = $4\text{ m}$
Rate of painting = Rs. 150 per $\text{m}^2$
Step 1: Calculate Curved Surface Area (CSA)
$$\text{CSA} = 2\pi r h$$
$$\text{CSA} = 2 \times \frac{22}{7} \times \frac{7}{10} \times 4$$
$$\text{CSA} = \frac{176}{10} = 17.6\text{ m}^2$$
Step 2: Calculate Total Cost
$$\text{Cost} = \text{CSA} \times \text{Rate}$$
$$\text{Cost} = 17.6 \times 150 = \text{Rs. } 2640$$
Answer: Total Cost = $$\text{Rs. } 2640$$
Q1: What is the formula for the curved surface area of a right circular cylinder?
Q2: If the radius of a cylinder is doubled while keeping height constant, the volume becomes:
Q3: How many liters are equivalent to $1\text{ m}^3$ of volume?
Q4: Find the total surface area of a cylinder with $r = 7\text{ cm}$ and $h = 7\text{ cm}$.
Q5: The volume of a cylinder with radius $r = 14\text{ cm}$ and height $h = 10\text{ cm}$ is: