Quantitative Aptitude

Mensuration

2,507 Questions
All Direct Formula Application 1067 Shape Transformations 343 Mensuration Ratio Equations 277 Combined & Hollow Solids 247 Successive Percentage Changes 178 Inscribed & Circumscribed Figures 165 Cost Estimation Problems 100 Liquid Pour & Submergence 67 Unit Conversions 35 Path & Border Calculations 17 Time & Work Integration 7 Scale & Model Proportions 2

Mensuration Questions

Multiple choice

A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1: 1 : 8: 27: 27. What is the respective ratio of the total surface area of given five cubes to the surface area of the formed cube? एक ठोस धात्विक घन को पाँच ठोस घन बनाने के लिए पिघलाया जाता है, जिसकी मात्रा 1: 1: 8: 27: 27 के अनुपात में होती है। दिए गए पाँच घनों के कुल प्रष्ठीय क्षेत्रफल और बने घन के प्रष्ठीय क्षेत्रफल का अनुपात क्या है?

  1. 3 : 4

  2. 4 : 3

  3. 3 : 2

  4. 2 : 3

  5. None of these इनमे से कोई नहीं
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the original cube have side 'a'. Volume = a³. The five new cubes have volumes in ratio 1:1:8:27:27, so their sides are 1:1:2:3:3 times a common factor. Total volume conserved: a³ = k³(1 + 1 + 8 + 27 + 27) = 64k³, so a = 4k. Surface area of original cube = 6a² = 96k². Surface areas of five cubes = 6k²(1 + 1 + 4 + 9 + 9) = 144k². Ratio = 144:96 = 3:2.

Multiple choice

Radius of a cone is increased by 20% and its height is reduced by 10%. Find the percentage change in its volume. एक शंकु के त्रिज्या में 20% की वृद्धि की गयी और इसकी ऊंचाई में 10% की कमी की गयी । इसके आयतन में प्रतिशत परिवर्तन ज्ञात कीजिये।

  1. increased by 29.6% 29.6% की वृद्धि

  2. reduced by 29.6% 29.6% की कमी

  3. increased by 32.6% 32.6% की वृद्धि

  4. reduced by 32.6% 32.6% की कमी

  5. None of these इनमे से कोई नहीं
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of cone V = (1/3)πr²h. New radius = 1.2r (20% increase). New height = 0.9h (10% decrease). New volume V' = (1/3)π(1.2r)²(0.9h) = (1/3)π × 1.44r² × 0.9h = 1.296 × (1/3)πr²h = 1.296V. Percentage change = (1.296V - V)/V × 100 = 0.296 × 100 = 29.6% increase.

Multiple choice

A hollow hemi-spherical vessel of steel of 0.5 cm thick and twice of its internal radius is 24cm. Then find the value of total surface area (in cm2) of the vessel?

  1. 600.2π

  2. 612.75π

  3. 600.5π

  4. 468.75π
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Internal radius r = 12 cm, thickness = 0.5 cm, so external radius R = 12.5 cm. For hollow hemisphere: Total surface area = Outer curved surface + Inner curved surface + Ring area at top = 2πR² + 2πr² + π(R² - r²) = 2π(12.5)² + 2π(12)² + π((12.5)² - 12²) = 2π(156.25) + 2π(144) + π(156.25 - 144) = 312.5π + 288π + 12.25π = 612.75π cm².

Multiple choice

The base radius of a solid cylinder is 12 cm and its height is 15 cm. If this solid cylinder is to be melted and recast into ‘n’ number of toys in the form of right circular cone , whose height is 9 cm and it is mounted on a hemisphere of radius 3 cm. Then find the value of ‘n’?

  1. 54

  2. 27

  3. 64

  4. 48
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

When the solid is melted and recast, volume remains constant. Original cylinder volume = π × 12² × 15 = 2160π. Each toy consists of a cone mounted on a hemisphere. Cone volume = (1/3)π × 3² × 9 = 27π. Hemisphere volume = (2/3)π × 3³ = 18π. Total toy volume = 45π. Number of toys n = 2160π/45π = 48.

Multiple choice

The base of a right prism is triangle whose sides are 5 cm, 8 cm and 12 cm. Find the lateral surface area of the right prism, if it’s height is 18 cm?

  1. 432

  2. 486

  3. 468

  4. 450
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a right prism with triangular base, lateral surface area = perimeter of base × height of prism. The base is a triangle with sides 5, 8, 12 cm. Perimeter = 5 + 8 + 12 = 25 cm. Height of prism = 18 cm. Lateral surface area = 25 × 18 = 450 cm².

Multiple choice

If the curved surface area of a right circular cone is 816.4 cm2 and the ratio between its radius and height is 5:12, then find the volume(in cm3) of the cone? ?(Take π=3.14)

  1. 31440

  2. 1256

  3. 2512

  4. 628
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Curved surface area of cone = πrl = 816.4, where r is radius and l is slant height. Given r/h = 5/12, so h = (12/5)r. For a cone, l² = r² + h² = r² + (144/25)r² = (169/25)r², so l = (13/5)r. Substituting: π × r × (13/5)r = 816.4. With π = 3.14: 3.14 × (13/5) × r² = 816.4. This gives r² = 100, so r = 10 cm. Then h = 24 cm. Volume = (1/3)πr²h = (1/3) × 3.14 × 100 × 24 = 2512 cm³.

Multiple choice

Praveen has a solid hemisphere of metal, he cut out a sphere of maximum volume. Then find the ratio of the volume of the sphere to that of the remaining solid ?

  1. 1 : 1

  2. 1 : 2

  3. 1 : 3

  4. 1 : 4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a hemisphere of radius R, the maximum sphere that can be cut has radius r = R/2 (inscribed in hemisphere). Volume of sphere = (4/3)π(R/2)³ = (4/3)πR³/8 = (1/6)πR³. Volume of hemisphere = (2/3)πR³. Remaining volume = (2/3)πR³ - (1/6)πR³ = (1/2)πR³. Ratio of sphere to remaining = (1/6)πR³ : (1/2)πR³ = 1:3. This matches option C.

Multiple choice

What is the capacity of the cylindrical tank? बेलनाकार टैंक की क्षमता क्या है? I. Radius of the base is half of its height. II. Area of the base is 462 sq. m. III. Height of the cylinder is 14 m. I. आधार का त्रिज्या इसकी ऊंचाई का आधा है। II. आधार का क्षेत्रफल 462 वर्ग मीटर है। III. बेलन की ऊंचाई 14 मीटर है।

  1. I and II I और II

  2. II and III II और III

  3. I and III I और III

  4. All I, II and III सभी I, II और III

  5. Any two are sufficient तीन में से कोई दो
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Cylinder capacity = πr²h. Statement II gives πr² = 462, so base area. Statement III gives h = 14. Together: capacity = 462 × 14 = 6468 m³. Statement I (r = h/2) with II gives: πr² = 462, so r = √(462/π), then h = 2r. With I and III: r = 14/2 = 7, capacity = π × 7² × 14. ANY pair gives enough info, making E correct.

Multiple choice

The radius of a circle is 20% more than the height of a right angled triangle. The base of triangle is 36 cm. If the area of triangle and circle be equal, what will be the area of circle? (Approx.) एक वृत्त की त्रिज्या एक समकोण त्रिभुज की ऊँचाई से 20% अधिक है। त्रिभुज का आधार 36 सेमी. है। यदि त्रिभुज और वृत्त का क्षेत्रफल बराबर हो तो वृत्त का क्षेत्रफल कितना है? (लगभग)

  1. 72 cm2.

  2. 144 cm2.

  3. 216 cm2.

  4. 128 cm2.

  5. 77 cm2.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Triangle area = (1/2) × base × height = (1/2) × 36 × h = 18h. Circle radius = 1.2h (20% more). Circle area = π × (1.2h)² = π × 1.44h² = 1.44πh². Equating areas: 1.44πh² = 18h, giving h = 18/(1.44π) ≈ 3.98. Circle area ≈ 1.44π × (3.98)² ≈ 71.8 cm², which rounds to 72 cm².

Multiple choice

The ratio of curved surface area of a right circular cylinder to the total area of its two bases is 2:1. If the total surface area of cylinder is 23100 square cm then what is the volume of cylinder?

  1. 247200

  2. 269500

  3. 312500

  4. 341800
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Curved surface area = 2πrh, total base area = 2πr². Given 2πrh : 2πr² = 2:1, so h : r = 2:1, meaning h = 2r. Total surface area = 2πrh + 2πr² = 2πr(2r) + 2πr² = 4πr² + 2πr² = 6πr² = 23100. So πr² = 23100/6 = 3850. Volume = πr²h = πr²(2r) = 2πr²×r = 2×3850×√(3850/π). Wait, let me solve directly. 6πr² = 23100, r² = 23100/(6π) = 3850/π. r = √(3850/π). h = 2r. Volume = πr²h = π×(3850/π)×2√(3850/π) = 3850×2√(3850/π) = 7700×√(3850/π). Using π ≈ 3.1416: √(3850/3.1416) ≈ √1225.54 ≈ 35.01. Volume ≈ 7700×35.01 ≈ 269577. Closest to 269500. Yes, option B.

Multiple choice

Three solid metallic sphere whose radii are 3cm, (x) cm and 18cm respectively, are melted and recast into a single solid sphere of diameter 38cm. Then find the total surface area of the sphere with radius (x) cm?
  1. $100\pi$
  2. $200\pi$
  3. $300\pi$
  4. $400\pi$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

When spheres are melted and recast, total volume remains constant. Volume of sphere = (4/3)πr³. Initial volume = (4/3)π(3³ + x³ + 18³) = (4/3)π(27 + x³ + 5832) = (4/3)π(x³ + 5859). Final sphere has diameter 38 cm, so radius = 19 cm. Its volume = (4/3)π(19)³ = (4/3)π(6859). Equating: x³ + 5859 = 6859, so x³ = 1000, x = 10 cm. Total surface area of sphere with radius x = 4πx² = 4π(100) = 400π, confirming option D.

Multiple choice

The lateral surface area of a cylinder is 792cm2. If its height is 14cm, then its volume (in cm3) is? [Take π=22/7]

  1. 1243

  2. 3564

  3. 2564

  4. 3929
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Lateral surface area of cylinder = 2πrh. Given LSA = 792 cm² and h = 14 cm: 2πr(14) = 792. Using π = 22/7: 2 × (22/7) × r × 14 = 792. Simplifying: 2 × 22 × r × 2 = 792, so 88r = 792, r = 9 cm. Volume = πr²h = (22/7) × 81 × 14 = 22 × 81 × 2 = 3564 cm³. This confirms option B is correct.

Multiple choice

A circle is drawn In ∆ABC, such that it touches sides AB, BC and AC at the points P,Q and R respectively. If side AB is 4 cm more than side BC, and side AB is 2 cm more than side AC , and the perimeter of ∆ABC=32 cm, then,find the value of BP+RA?

  1. 12 cm

  2. 13 cm

  3. 33/5 cm

  4. 38/3 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let BC = x. Then AB = x + 4 and AC = x + 2. Perimeter: x + (x+4) + (x+2) = 32, giving 3x = 26, so x = 26/3 = BC. By equal tangents: BP = BQ, RA = RC, and CP = CQ. BP + RA = BQ + RC. From tangency: BQ + CP = BC and CP + CR = AC. After calculation: BP + RA = 38/3 cm.