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Cylinder, Cone, and Sphere

Introduction

Cylinders, cones, and spheres are fundamental 3D solids in mensuration that often appear in practical problems involving containers, pipes, balls, and cones. Understanding how to calculate their surface areas and volumes is essential for both real-world applications and aptitude exams.

These shapes are all derived from circles, so their formulas heavily use π (pi). This pattern teaches you how to handle curved surfaces and volume relationships efficiently.

Pattern: Cylinder, Cone, and Sphere

Pattern

The key idea is to use the correct formula for each solid depending on whether you are asked for surface area, curved surface area, or volume.

Formulas to Remember:

Cylinder:
• Volume = πr²h
• Curved Surface Area (CSA) = 2πrh
• Total Surface Area (TSA) = 2πr(h + r)

Cone:
• Volume = (1/3)πr²h
• Slant height (l) = √(r² + h²)
• CSA = πrl
• TSA = πr(l + r)

Sphere:
• Volume = (4/3)πr³
• Surface Area = 4πr²

Step-by-Step Example

Question

A cone has a radius of 7 cm and height of 24 cm. Find its slant height and curved surface area (CSA).

Solution

  1. Step 1: Find the slant height (l).

    Using Pythagoras theorem: l = √(r² + h²) = √(7² + 24²) = √(49 + 576) = √625 = 25 cm.

  2. Step 2: Find the curved surface area (CSA).

    CSA = πrl = (22/7) × 7 × 25 = 22 × 25 = 550 cm².

  3. Final Answers:

    • Slant height = 25 cm
    • CSA = 550 cm²

  4. Quick Check:

    Substituting values again confirms: (22/7) × 7 × 25 = 550 ✅

Quick Variations

1. Find radius or height when volume or surface area is given.

2. Compare volumes of cone and cylinder having the same base and height (ratio 1:3).

3. Use hemispheres (half-sphere) in tank or dome-based problems.

4. Combine solids like a cone on a cylinder or a sphere inside a cylinder.

Trick to Always Use

  • Step 1 → Identify the solid type (cylinder, cone, or sphere).
  • Step 2 → Write down its formula carefully before substituting.
  • Step 3 → For cones, always compute slant height first using √(r² + h²).
  • Step 4 → Use π = 22/7 unless specified otherwise.

Summary

Summary

In the Cylinder, Cone, and Sphere pattern:

  • Cylinder: Volume = πr²h; TSA = 2πr(h + r)
  • Cone: Volume = (1/3)πr²h; Slant height = √(r² + h²)
  • Sphere: Volume = (4/3)πr³; Surface Area = 4πr²
  • Always check units - area in cm², volume in cm³.
  • For hemisphere → halve the sphere’s volume and add base area if required.

Practice

(1/5)
1. Find the volume of a cylinder with radius 7 cm and height 10 cm (Take π = 22/7).
easy
A. 1540 cm³
B. 1500 cm³
C. 1600 cm³
D. 1400 cm³

Solution

  1. Step 1: Formula.

    Volume of cylinder = πr²h.

  2. Step 2: Substitute values.

    Volume = (22/7) × 7² × 10 = (22/7) × 49 × 10.

  3. Step 3: Simplify.

    49/7 = 7 so Volume = 22 × 7 × 10 = 1540 cm³.

  4. Final Answer:

    1540 cm³ → Option A.

  5. Quick Check:

    π×49×10 with π=22/7 gives 22×70 = 1540 ✅
Hint: Square the radius first, then multiply by π and height.
Common Mistakes: Forgetting to square radius or omitting π.
2. A cone has a radius of 3.5 cm and height 10.5 cm. Find its volume. (Use π = 22/7)
easy
A. 138.60 cm³
B. 134.75 cm³
C. 135.25 cm³
D. 140.00 cm³

Solution

  1. Step 1: Formula.

    Volume of cone = (1/3)πr²h.

  2. Step 2: Substitute values.

    r = 3.5 = 7/2 so r² = 49/4; h = 10.5 = 21/2. Volume = (1/3)×(22/7)×(49/4)×(21/2).

  3. Step 3: Simplify.

    49/7 = 7, so Volume = (1/3)×22×7×21 / (4×2) = (1/3)×22×7×21/8 = (22×7×21)/(24) = 3234/24 = 134.75 cm³.

  4. Final Answer:

    134.75 cm³ → Option B.

  5. Quick Check:

    Compute r²h first (12.25×10.5=128.625); (π/3)×128.625 ≈ (22/21)×128.625 ≈ 134.75 ✅
Hint: Compute πr²h then divide by 3; keep fractions to simplify cancellations.
Common Mistakes: Omitting the 1/3 factor or using diameter instead of radius.
3. Find the curved surface area (CSA) of a cylinder of radius 14 cm and height 20 cm. (π = 22/7)
easy
A. 1680 cm²
B. 1600 cm²
C. 1760 cm²
D. 1840 cm²

Solution

  1. Step 1: Formula.

    CSA of cylinder = 2πrh.

  2. Step 2: Substitute values.

    CSA = 2 × (22/7) × 14 × 20.

  3. Step 3: Simplify.

    14/7 = 2 so CSA = 2 × 22 × 2 × 20 = 1760 → 1760 cm².

  4. Final Answer:

    1760 cm² → Option C.

  5. Quick Check:

    2πrh = 2×22/7×14×20 = 2×22×40 = 1760 ✅
Hint: Cancel 7 with 14 early: (14/7 = 2) to simplify mentally.
Common Mistakes: Using volume formula instead of CSA.
4. Find the total surface area (TSA) of a sphere with radius 7 cm (π = 22/7).
medium
A. 600 cm²
B. 650 cm²
C. 700 cm²
D. 616 cm²

Solution

  1. Step 1: Formula.

    Surface area of sphere = 4πr².

  2. Step 2: Substitute values.

    TSA = 4 × (22/7) × 7² = 4 × (22/7) × 49.

  3. Step 3: Simplify.

    49/7 = 7 so TSA = 4 × 22 × 7 = 616 cm².

  4. Final Answer:

    616 cm² → Option D.

  5. Quick Check:

    4πr² with π=22/7 gives 4×22×7 = 616 ✅
Hint: Cancel 7 with r² early to speed calculation.
Common Mistakes: Mixing up sphere surface area with cylinder formulas.
5. A cone has slant height 13 cm and radius 5 cm. Find its curved surface area (π = 22/7).
medium
A. 204.29 cm²
B. 200.19 cm²
C. 220.10 cm²
D. 245.00 cm²

Solution

  1. Step 1: Formula.

    Curved surface area (CSA) of cone = π × r × l.

  2. Step 2: Substitute values.

    CSA = (22/7) × 5 × 13 = 1430 ÷ 7 = 204.2857…

  3. Step 3: Round sensibly.

    CSA ≈ 204.29 cm².

  4. Final Answer:

    204.29 cm² → Option A.

  5. Quick Check:

    π×5×13 ≈ 3.1416×65 ≈ 204.2 - matches using 22/7 ✅
Hint: Multiply radius by slant height first (r×l), then multiply by π.
Common Mistakes: Using vertical height instead of slant height in CSA formula.

Mock Test

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