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Mensuration: Surface Area and Volume - Compute curved and total surface area and volume of cylinders

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A cylinder is a solid with two congruent circular bases connected by a curved surface. The distance between the centers of the circular bases is the height hh and the radius of the circular base is rr.

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The Curved Surface Area (CSA) of a cylinder is the area of the side surface, excluding the top and bottom. It can be visualized as unrolling a rectangle with length equal to the circumference 2πr2\pi r and width equal to the height hh.

Unrolled curved surface of a cylinder shown as a rectangle
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The Total Surface Area (TSA) of a cylinder is the sum of the curved surface area and the areas of the two circular bases: TSA=2πrh+2πr2=2πr(r+h)TSA = 2\pi rh + 2\pi r^2 = 2\pi r(r + h).

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The volume of a cylinder represents the amount of space it occupies, calculated by multiplying the area of the base πr2\pi r^2 by the height hh.

📐Formulae

Area of the circular base = πr2\pi r^2

Volume of a Cylinder (VV) = πr2h\pi r^2 h

Radius (rr) in terms of Diameter (dd) = d2\frac{d}{2}

Height (hh) = Vπr2\frac{V}{\pi r^2}

Capacity in Liters (if VV is in cm3cm^3) = V1000\frac{V}{1000}

Capacity in Liters (if VV is in m3m^3) = V×1000V \times 1000

💡Examples

Problem 1:

Find the volume of a right circular cylinder with a base radius of 7cm7 cm and a height of 10cm10 cm. (Use π=227\pi = \frac{22}{7})

Solution:

Given: Radius (rr) = 7cm7 cm, Height (hh) = 10cm10 cm. Using the formula: V=πr2hV = \pi r^2 h V=227×7×7×10V = \frac{22}{7} \times 7 \times 7 \times 10 V=22×7×10V = 22 \times 7 \times 10 V=154×10V = 154 \times 10 V=1540cm3V = 1540 cm^3

Explanation:

To find the volume, we substitute the given radius and height into the volume formula. The value of π\pi is taken as 227\frac{22}{7} to simplify calculation with the radius of 77. The final result is expressed in cubic centimeters.

Problem 2:

The capacity of a closed cylindrical vessel of height 1m1 m is 15.4liters15.4 liters. How many square meters of metal sheet would be needed to make it? (Find the radius first to determine volume components)

Solution:

Given: Capacity = 15.4liters15.4 liters, Height (hh) = 1m1 m. First, convert capacity to volume in m3m^3: 15.4liters=15.41000m3=0.0154m315.4 liters = \frac{15.4}{1000} m^3 = 0.0154 m^3 Now, use the volume formula to find radius (rr): V=πr2hV = \pi r^2 h 0.0154=227×r2×10.0154 = \frac{22}{7} \times r^2 \times 1 r2=0.0154×722r^2 = \frac{0.0154 \times 7}{22} r2=0.0007×7=0.0049r^2 = 0.0007 \times 7 = 0.0049 r=0.0049=0.07mr = \sqrt{0.0049} = 0.07 m

Explanation:

First, we convert the capacity from liters to m3m^3 since the height is in meters. We then use the volume formula V=πr2hV = \pi r^2 h to solve for the unknown radius rr. This radius is crucial for any further calculations regarding the cylinder's dimensions.

Problem 3:

A soft drink is available in a cylindrical tin with a plastic base of diameter 7 cm7 \text{ cm} and height 10 cm10 \text{ cm}. Find the capacity of the tin container. (Use π=227\pi = \frac{22}{7})

Cylindrical tin with radius 3.5 cm and height 10 cm

Solution:

  1. Identify Given Values: Diameter (dd) = 7 cm7 \text{ cm} Radius (rr) = d2=72=3.5 cm\frac{d}{2} = \frac{7}{2} = 3.5 \text{ cm} Height (hh) = 10 cm10 \text{ cm}

  2. Apply Volume Formula: V=πr2hV = \pi r^2 h V=227×3.5×3.5×10V = \frac{22}{7} \times 3.5 \times 3.5 \times 10 V=227×72×72×10V = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 10

  3. Calculation: V=11×7×5V = 11 \times 7 \times 5 V=385 cm3V = 385 \text{ cm}^3

  4. Conclusion: The capacity of the tin container is 385 cm3385 \text{ cm}^3.

Explanation:

The capacity of a container is equivalent to its internal volume. Since the tin is cylindrical, we use the formula for the volume of a cylinder. Note that radius is half of the diameter.

Problem 4:

The curved surface area of a right circular cylinder of height 14 cm14 \text{ cm} is 88 cm288 \text{ cm}^2. Find the diameter of the base of the cylinder.

Cylinder with height 14 cm and unknown diameter

Solution:

  1. Identify Given Values: Curved Surface Area (CSACSA) = 88 cm288 \text{ cm}^2 Height (hh) = 14 cm14 \text{ cm}

  2. Apply CSA Formula: CSA=2πrhCSA = 2 \pi r h 88=2×227×r×1488 = 2 \times \frac{22}{7} \times r \times 14

  3. Solve for r: 88=44×2×r88 = 44 \times 2 \times r 88=88×r88 = 88 \times r r=8888=1 cmr = \frac{88}{88} = 1 \text{ cm}

  4. Find Diameter: d=2×rd = 2 \times r d=2×1=2 cmd = 2 \times 1 = 2 \text{ cm}

  5. Conclusion: The diameter of the base of the cylinder is 2 cm2 \text{ cm}.

Explanation:

The Curved Surface Area (CSA) is the area of the side surface. By substituting the known CSA and height into the formula 2πrh2 \pi r h, we solve for the unknown radius rr, then double it to find the diameter.