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Geometry / Mensuration Based Data Sufficiency

Introduction

Geometry and Mensuration based Data Sufficiency problems test your ability to determine whether the provided information is enough to find measurements such as area, perimeter, volume, or dimensions of geometric figures. These questions are not about calculating the value, but rather checking if the statements give enough data to determine it uniquely.

This pattern is vital because it blends geometric reasoning with logical sufficiency testing - a key component in reasoning and aptitude exams.

Pattern: Geometry / Mensuration Based Data Sufficiency

Pattern

The key idea is to use geometric or mensuration formulas (like Area = L × B, Circumference = 2πr, etc.) and test if one or both statements give enough data to compute the required dimension.

You should never find the actual numeric value; the goal is to decide whether each statement provides enough independent or combined information to determine the answer.

Step-by-Step Example

Question

What is the area of a rectangle?
(I) Length = 8 cm.
(II) Perimeter = 24 cm.

Choose the correct option:
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze Statement (I)

    Length = 8 cm, but breadth is not known → (I) alone is insufficient.

  2. Step 2: Analyze Statement (II)

    Perimeter = 24 cm ⇒ 2(L + B) = 24 ⇒ L + B = 12 → still two variables → (II) alone is insufficient.

  3. Step 3: Combine Statements

    From (I): L = 8; substitute in (II): 8 + B = 12 ⇒ B = 4. Now, Area = L × B = 8 × 4 = 32 cm² → both statements together are sufficient.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Length 8, Breadth 4 → Area = 32 ✅

Quick Variations

1. Questions based on area, perimeter, and diagonal of rectangle or square.

2. Questions involving radius, circumference, or area of circle.

3. Volume and surface area of cubes, cuboids, and cylinders.

4. Triangular geometry using base, height, or sides (Heron’s formula).

5. Mixed-figure sufficiency (e.g., a square inscribed in a circle).

Trick to Always Use

  • Step 1: Write down the required formula (Area, Volume, etc.).
  • Step 2: Check if the statement provides all required variables for that formula.
  • Step 3: Test each statement independently before combining.
  • Step 4: Combine only if neither alone provides complete data.

Summary

Summary

  • Geometry DS problems rely on formulas and complete variable sets, not actual calculations.
  • One statement is sufficient only if it provides all required dimensions.
  • If each gives partial data, both together may be needed.
  • Always test sufficiency - not compute actual measurement.

Example to remember:
(I) Length = 8 cm; (II) Perimeter = 24 cm → Both together sufficient to find Area = 32 cm².

Practice

(1/5)
1. What is the area of a square plot?<br>(I) The perimeter of the square is 48 m.<br>(II) The diagonal of the square is 16√2 m.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze Statement (I)

    Perimeter = 48 ⇒ Side = 48 ÷ 4 = 12 m → Area = 12² = 144 m² → (I) alone sufficient.

  2. Step 2: Analyze Statement (II)

    Diagonal = 16√2 ⇒ Side = (16√2) ÷ √2 = 16 m → Area = 16² = 256 m² → (II) alone sufficient.

  3. Final Answer:

    Each statement alone is sufficient → Option C

  4. Quick Check:

    Both statements independently provide full data ✅
Hint: For a square, one dimension (side, perimeter, or diagonal) is enough to find area.
Common Mistakes: Thinking both statements are required when one already defines the square.
2. What is the radius of a circle?<br>(I) The circumference of the circle is 44 cm.<br>(II) The area of the circle is 154 cm².
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    Circumference = 2πr = 44 ⇒ r = 44 ÷ (2×3.14) ≈ 7 cm → (I) alone sufficient.

  2. Step 2: Analyze (II)

    Area = πr² = 154 ⇒ r² = 154 ÷ 3.14 ⇒ r ≈ 7 cm → (II) alone sufficient.

  3. Final Answer:

    Each statement alone is sufficient → Option C

  4. Quick Check:

    Both give r = 7 ✅
Hint: For circles, knowing either circumference or area gives radius directly.
Common Mistakes: Using both statements when one is enough to compute radius.
3. What is the volume of a cylinder?<br>(I) The height of the cylinder is 10 cm.<br>(II) The radius of the cylinder is 7 cm.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    Height = 10 cm, radius unknown → insufficient.

  2. Step 2: Analyze (II)

    Radius = 7 cm, height unknown → insufficient.

  3. Step 3: Combine

    Volume = πr²h = 3.14×7²×10 = 1539 cm³ → both needed.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Volume calculable only when both radius & height known ✅
Hint: For 3D solids, all required dimensions must be known to compute volume.
Common Mistakes: Assuming one dimension defines a solid’s volume.
4. What is the area of a triangle?<br>(I) Base = 10 cm.<br>(II) Height = 12 cm.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Evaluate (I)

    Base known, height missing → insufficient.

  2. Step 2: Evaluate (II)

    Height known, base missing → insufficient.

  3. Step 3: Combine

    Area = ½ × 10 × 12 = 60 cm² → both together sufficient.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Both required variables available only when combined ✅
Hint: For triangle area, both base and height must be known.
Common Mistakes: Assuming either base or height alone determines area.
5. What is the total surface area of a cube?<br>(I) The length of each edge is 5 cm.<br>(II) The volume of the cube is 125 cm³.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    Edge = 5 ⇒ Surface Area = 6 × 5² = 150 cm² → sufficient.

  2. Step 2: Analyze (II)

    Volume = 125 ⇒ Edge = ∛125 = 5 ⇒ same result → sufficient.

  3. Final Answer:

    Each statement alone is sufficient → Option C

  4. Quick Check:

    Both statements lead to same surface area ✅
Hint: For cubes, knowing either edge or volume gives every measurement.
Common Mistakes: Combining statements unnecessarily when one fully defines the cube.

Mock Test

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