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

What is the area of the sector whose central angle is 90° and radius of the circle is 14 cm? उस सेक्टर का क्षेत्रफल क्या होगा जिसका केंद्रीय कोण 90° है और वृत्त की त्रिज्या 14 सेमी. है?

  1. 308 sq cm / वर्ग सेमी.

  2. 77 sq cm / वर्ग सेमी.

  3. 154 sq cm / वर्ग सेमी.

  4. 231 sq cm / वर्ग सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of sector = (θ/360°) × πr² = (90/360) × (22/7) × 14² = (1/4) × (22/7) × 196 = 154 sq cm. Option A (308) is the full circle area, B (77) is half the correct value, and D (231) is 3/4 of the circle area.

Multiple choice

The external and internal radii of a hollow right circular cylinder of height 15 cm are 6.75 cm and 5.25 cm respectively. If it is melted and recasted into a solid cylinder of height equal to half of the original cylinder, then the radius of solid cylinder is- 15 सेमी. ऊँचाई के खोखले बेलन का बाह्य एव आन्तरिक त्रिज्याएं क्रमशः 6.75 सेमी. और 5.25 सेमी. है। यदि इसको गलाकर एक ठोस बेलन जिसकी ऊँचाई वास्तविक बेलन की ऊँचाई के आधे के बराबर है, तो ठोस बेलन की त्रिज्या है-

  1. 6

  2. 6.5

  3. 7.25

  4. 7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of hollow cylinder = πh(R² - r²) = π × 15(6.75² - 5.25²) = π × 15(45.5625 - 27.5625) = π × 15 × 18 = 270π. New solid cylinder has height = 7.5cm. Volume = π × 7.5 × R². Equating: π × 7.5 × R² = 270π. R² = 270/7.5 = 36. R = 6cm.

Multiple choice

A cylindrical iron pillar 49 cm. high and 6 cm. in radius is surmounted by a cone 14 cm. high. The volume of pillar is - एक बेलनाकार लोहे के खम्भे की ऊँचाई 49 सेमी. और अन्तः त्रिज्या 6 सेमी. है। इसके ऊपर 14 सेमी. ऊँचाई का एक शंकु लगाया गया है। खम्भे का आयतन है -

  1. 5981 cm.3/सेमी.3

  2. 7012 cm.3/सेमी.3

  3. 6072 cm.3/सेमी.3

  4. 8154 cm.3/सेमी.3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Volume of cylinder = πr²h = π × 6² × 49 = 1764π. Volume of cone = (1/3)πr²h = (1/3)π × 6² × 14 = 168π. Total volume = 1764π + 168π = 1932π ≈ 6072 cm³ (using π ≈ 22/7).

Multiple choice

If a metallic cone of radius 30 cm. and height 45 cm. is melted and recast into metallic sphere of radius 5cm., find the number of spheres. यदि 30 सेमी.त्रिज्या और 45 सेमी. ऊँचाई वाले एक शंकु को पिघलाकर 5 सेमी. त्रिज्या वाले गोलों में पुनः ढाला जाए तो गोलों की संख्या ज्ञात कीजिए।

  1. 405

  2. 91

  3. 80

  4. 81
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Volume of cone = (1/3)πr²h = (1/3)π(30)²(45) = (1/3)π(900)(45) = 13500π cm³. Volume of each sphere = (4/3)πr³ = (4/3)π(5)³ = (4/3)π(125) = 500π/3 cm³. Number of spheres = Volume of cone / Volume of sphere = 13500π / (500π/3) = 13500π × (3/500π) = 13500 × 3 / 500 = 81 spheres. The π cancels out and we get exactly 81.

Multiple choice

The radius of a metallic cylinder is 3 cm and its height is 5 cm. It is melted and moulded into small cones each of height 1 cm and base 1 mm. The number of such cones formed, is?. धात्विक बेलन की त्रिज्या 3 सेमी. और ऊँचाई 5 सेमी. है। यदि उसे गलाया जाये और उसे छोटे शंकुओं में जिसमें प्रत्येक की ऊँचाई 1 सेमी तथा आधार 1 मिमी है, में परिवर्तित किया जाये तो ऐसे कितने शंकु बनेंगे?

  1. 12500

  2. 13500

  3. 14500

  4. 15500
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Volume of cylinder = πr²h = π(3)²(5) = 45π cm³. Volume of each small cone = (1/3)πr²h = (1/3)π(0.1)²(1) = π/300 cm³ (note: 1 mm = 0.1 cm). Number of cones = cylinder volume / cone volume = 45π / (π/300) = 45 × 300 = 13500. The answer 13500 is correct.

Multiple choice

The curved surface area of a cylindrical rod is 264 m2. and its volume is 924 m3. The ratio of its semi diameter to height is - एक बेलनाकार छड़ का पार्श्व पृष्ठ क्षेत्रफल 264 मी.2 है और इसका आयतन 924 मी.3 है | इसके अर्द्ध व्यास का इसकी ऊँचाई से अनुपात है -

  1. 3 : 7

  2. 7 : 3

  3. 6 : 7

  4. 7 : 6
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

CSA = 2πrh = 264, Volume = πr²h = 924. Dividing: (πr²h)/(2πrh) = 924/264 = r/2 = 7/2. So r = 7. Semi-diameter (radius) : height = 7 : 6. This ratio can be derived from the formulas.

Multiple choice

The sum of length, breadth and height of a room is 19 m. The length of the diagonal is 11m. The cost of painting the total surface area of the room at the rate of Rs. 8 per square metre is- (in Rs.) एक कमरे की लम्बाई, चौड़ाई और ऊँचाई का योग 19 मी. है। विकर्ण की लम्बाई 11 मी. है। कमरे के कुल पृष्ठ क्षेत्रफल को 8 रू. प्रति वर्ग मी. की दर से पेन्ट कराने का खर्च है-(रू. में)

  1. 2400

  2. 1920

  3. 1620

  4. 1580
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given l+b+h=19 and diagonal² = l²+b²+h² = 121. Using (l+b+h)² = l²+b²+h²+2(lb+bh+lh), we get 361 = 121 + 2(lb+bh+lh). Therefore surface area = 2(lb+bh+lh) = 240 sq.m. Cost = 240 × 8 = Rs. 1920.

Multiple choice

From the four corners of a rectangular sheet of dimensions 25 cm x 20 cm, square of side 2 cm is cut off from four corners and a box is made. The volume of the box is- 25 सेमी. 20 सेमी आयताकार शीट के चारों कोनों से 2 सेमी भुजा के चार वर्ग काटकर एक बॉक्स बना दिया जाता है। बॉक्स का आयतन है।

  1. 828 cm3

  2. 672 cm3

  3. 500 cm3

  4. 1000cm3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

After cutting 2 cm squares from each corner, the new dimensions are: length = 25 - 2 - 2 = 21 cm, width = 20 - 2 - 2 = 16 cm, and height = 2 cm (the flaps). Volume = length × width × height = 21 × 16 × 2 = 672 cm³. This is a classic optimization/application problem where cutting and folding creates an open box.

Multiple choice

If the height, curved surface area and the volume of a cone are h, c and v respectively then- 3πvh3–c2h2+9v2 will be equal to- यदि एक शंकु की ऊँचाई, वक्रपृष्ठ क्षेत्रफल और आयतन h,c और v है तो 3πvh3–c2h2+9v2 बराबर होगा-

  1. –1

  2. 9V

  3. 1

  4. 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a cone, the expression 3πv-h² + 9v² - c²h² simplifies to 0 by substituting v = (1/3)πr²h and c = πrl, then using l² = r² + h². After algebraic manipulation, all terms cancel out, yielding 0. This tests the relationship between cone dimensions.

Multiple choice

A solid right circular cone is cut perpendicular to its height from base in ratio 2:3 Find the ratio of volume of upper half to lower half. एक ठोस लम्बवृत्तीय शंकु को ऊंचाई के लम्बवत आधार से 2: 3 के अनुपात में काटा जाता है ऊपरी आधे हिस्से तथा निचले आधे हिस्से के आयतन का अनुपात ज्ञात करें ।

  1. 27:64

  2. 27:98

  3. 64:125

  4. 27:125
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

When a cone is cut parallel to its base, the upper portion is a smaller similar cone. The height ratio is 2:3 from base, meaning the upper cone has height 3/5 of the original. For similar cones, volumes are proportional to the cube of the height ratio. Upper cone volume = (3/5)^3 = 27/125 of the original. Lower frustum volume = 1 - 27/125 = 98/125 of the original. Ratio of upper to lower = 27/125 : 98/125 = 27:98.

Multiple choice

Length of a rectangle is reduced by 25%, its breadth should be increased by how much percent so that its area increased by 2%? एक चतुर्भुज की लम्बाई 25% कम कर दी गयी इसकी चौड़ाई में कितने प्रतिशत की वद्धि करनी चाहिये ताकि इसका क्षेत्रफल 2 प्रतिशत बढ़ जाये ?

  1. 32%

  2. 27%

  3. 41.5%

  4. 36%
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let original length = L, breadth = B. New length = 0.75L (25% reduction). Let new breadth = B'. We need new area = 1.02LB (2% increase). So 0.75L × B' = 1.02LB. B' = 1.02B/0.75 = 1.36B. This means breadth increases by 36%.

Multiple choice

(\text{Volume of two circles is in the ratio of 8 : 27. Find the ratio of their curved surface area ?})
  1. $\(4 : 9\)$
  2. $\(9 : 4\)$
  3. $\(4 : 5\)$
  4. $\(2 : 3\)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume ratio 8:27 means the cubes of radii are in this ratio (for spheres). Taking cube root, the radii are in ratio 2:3. Curved surface area of a sphere is 4πr², so the surface areas are in the ratio of squares of radii: 2²:3² = 4:9. Note: The question incorrectly says 'circles' instead of 'spheres' - circles are 2D shapes and don't have volume.

Multiple choice

The ratio between the total surface area of a solid right circular cylinder and a solid sphere is 2 : 1. If they have the same radii, the ratio of the volume of the cylinder to that of the sphere is given by - एक ठोस लम्ब वृत्तीय बेलन और एक गोले के कुल पृष्ठीय क्षेत्रफल में अनुपात 2 : 1 है। यदि उनकी त्रिज्यायें समान हैं, तो बेलन के आयतन का गोले के आयतन से अनुपात है-

  1. 9 : 4

  2. 2 : 1

  3. 3 : 4

  4. 4 : 9
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a cylinder with radius r and height h: total surface area = 2πr² + 2πrh, volume = πr²h. For a sphere with radius r: surface area = 4πr², volume = (4/3)πr³. Given surface area ratio = 2:1, we have (2πr² + 2πrh) / (4πr²) = 2/1. This simplifies to (2 + 2h/r) / 4 = 2, giving 2 + 2h/r = 8, so h/r = 3. The volume ratio = (πr²h) / ((4/3)πr³) = h / (4r/3) = 3h / 4r. Substituting h = 3r: volume ratio = 3(3r) / 4r = 9/4.