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

If the radius of a sphere is increased by 2 cm. Its surface area increased by 352 cm2. The radius of sphere before change is यदि एक गोले की त्रिज्या में 2 सेमी. की वृद्धि की जाती है | इसका पृष्ठ क्षेत्रफल 352 सेमी.2 से बढ़ जाता है | परिवर्तन से पहले गोले की त्रिज्या है

  1. 3 cm./सेमी.

  2. 4 cm./सेमी.

  3. 5 cm./सेमी.

  4. 6 cm./सेमी.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the original radius be r cm. When increased by 2 cm, new radius is (r+2) cm. Surface area of sphere = 4πr². The increase in surface area is 4π(r+2)² - 4πr² = 352. Solving: 4π(r²+4r+4-r²) = 352 → 4π(4r+4) = 352 → 16π(r+1) = 352 → π(r+1) = 22. Using π = 22/7: r+1 = 7, so r = 6 cm. This matches option D.

Multiple choice

A hollow sphere of external and internal radius 8 cm and 6 cm respectively is melted into a cone of base diameter 12 cm.Find the approx.height of the cone. बाहरी और भीतरी त्रिज्याएँ क्रमश : 8 सेमी और 6 सेमी वाले एक खोखले गोले को 12 सेमी व्यास वाले शंकु के रूप में पिघलाया जाता है | शंकु की लगभग ऊँचाई ज्ञात कीजिए |

  1. 21

  2. 33

  3. 37

  4. 41
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Volume of hollow sphere = (4/3)π(8³ - 6³) = (4/3)π(512 - 216) = (4/3)π(296) ≈ 1239.9 cm³. Cone: base diameter = 12 cm, so radius = 6 cm. Volume = (1/3)πr²h = (1/3)π(36)h = 12πh. Equating: 12πh = 1239.9, so h = 1239.9/(12π) ≈ 32.8 cm ≈ 33 cm.

Multiple choice

The side of the first square exceeds the side of another square by 4 cm. and the sum of the areas of the two squares is 400 sq. cm. Find the dimensions of the squares. पहले वर्ग की भुजा दूसरे वर्ग की भुजा से 4 सेमी. अधिक है, और इनके क्षेत्रफलों का योग 400 वर्ग सेमी. है | वर्गों की विमायें ज्ञात कीजिये |

  1. 12, 16

  2. 16, 20

  3. 12, 18

  4. 12, 14
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the smaller side be x cm. Then x² + (x+4)² = 400. Expanding: x² + x² + 8x + 16 = 400, so 2x² + 8x - 384 = 0. Dividing by 2: x² + 4x - 192 = 0. Using the quadratic formula: x = [-4 ± √(16 + 768)]/2 = [-4 ± √784]/2 = [-4 ± 28]/2. Taking the positive value: x = 12 cm. The other square has side x+4 = 16 cm. These satisfy the conditions: 12² + 16² = 144 + 256 = 400.

Multiple choice

If the height of a cylinder is increased by 20% and the radius of the base is decreased by 10% then by what percent will its curved surface area change? यदि एक बेलन की ऊँचाई में 20% वृद्धि कर दी जाती है और आधार की त्रिज्या में 10% की कमी कर दी जाती है, तो इसके वक्र पृष्ठीय क्षेत्रफल में कितने प्रतिशत का बदलाव आयेगा?

  1. 4%

  2. 6%

  3. 8%

  4. 11%
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Original curved surface area = 2πrh. New CSA = 2π(0.9r)(1.2h) = 2πrh × 1.08. This represents an 8% increase from the original.

Multiple choice

The total surface area of a solid hemisphere is 108π cm2. Find the volume of the hemisphere. एक अर्द्ध गोले का कुल पृष्ठीय क्षेत्रफल 108π सेमी2. है। अर्द्धगोले का आयतन ज्ञात कीजिए।

  1. 72π cm3.

  2. 144π cm3.

  3. 108 cm3.

  4. 54 cm3.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Total surface area of hemisphere = 3πr² = 108π, so r² = 36 and r = 6 cm. Volume of hemisphere = (2/3)πr³ = (2/3)π(216) = 144π cm³. The key is remembering the total surface area includes the circular base (3πr² not 2πr²).

Multiple choice

The questions given below contain three statements giving certain data. You have to decide whether the data given in the statements are sufficient for answering the question? What will be the increase percent in volume? I: An ice cube is dipped into 15 litre water II: If radius of a cylinder increases by 20% and height increases by 30% III: If the height of ice cube is 10 cm.

  1. Only I and II

  2. Only II and III

  3. II or (I and III)

  4. Any of two statements

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

Statement II gives complete information for a cylinder: radius increases by 20%, height by 30%. New volume = π(1.2r)²(1.3h) = 1.872πr²h, which is an 87.2% increase. Statements I and III together describe an ice cube scenario. The question ambiguously doesn't specify the object, so either path works.

Multiple choice

How many spheres of diameter 2 cm can be made by melting a sphere of radius 4 cm? 4 सेमी की त्रिज्या वाले एक गोले को पिघलाकर 2 सेमी वाले व्यास के कितने गोले बनाये जा सकते हैं?

  1. 64

  2. 16

  3. 32

  4. 128
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Volume of original sphere = (4/3)π(4)³ = (4/3)π × 64 = 256π/3. Volume of each small sphere = (4/3)π(1)³ = 4π/3 (radius is 1 cm since diameter is 2 cm). Number of spheres = (256π/3) / (4π/3) = 256/4 = 64. Therefore, option A is correct. Volume is conserved when melting and recasting.

Multiple choice

If the ratio of Area and diameter of a circle is 35 : 14 then find the circumference of the circle. यदि वृत्त के क्षेत्रफल और उसके व्यास का अनुपात 35 : 14 है तो वृत्त की परिधि ज्ञात कीजिए।

  1. 7

  2. 10

  3. 12

  4. 18
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area = πr², diameter = 2r. Given πr² / 2r = 35/14, so πr/2 = 35/14, which gives r = 5/π. Circumference = 2πr = 2π × (5/π) = 10. The π terms cancel out conveniently.

Multiple choice

A rectangular piece of cardboard 18 cm \times 24 cm is made into an open box by cutting a square of 5 cm side from each corner and building up the side. What is the volume of the box? (in cm$^3$)
  1. $1235$
  2. $990$
  3. $560$
  4. $600$
  5. $Can’t be determined$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Original cardboard: 18 x 24 cm. After cutting 5 cm squares from each corner, the new dimensions of the box base are: length = 24 - 5 - 5 = 14 cm, width = 18 - 5 - 5 = 8 cm, height = 5 cm (the folded-up sides). Volume = length x width x height = 14 x 8 x 5 = 560 cm³. The key insight is that cutting squares from corners reduces both dimensions by twice the square's side length.

Multiple choice

If a wire of length 36 cm. is bend in a shape of semi circle then find the area of the semi circle. (in cm2) यदि 36 सेमी. लम्बाई वाले तार को मोड़कर अर्द्धवृत्त का आकार दिया जाये तो अर्द्धवृत्त का क्षेत्रफल ज्ञात कीजिए। (सेमी.2 में)

  1. 77 cm2/सेमी2

  2. 72 cm2/सेमी2

  3. 82 cm2/सेमी2

  4. 144 cm2/सेमी2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Semi-circle perimeter includes half-circumference + diameter. Using π = 22/7: perimeter = πr + 2r = 36, so (22/7)r + 2r = 36 → r(22/7 + 2) = 36 → r = 7. Area = (πr²)/2 = (22/7 × 49)/2 = 77 cm². The claimed answer is correct assuming π = 22/7.

Multiple choice

A cube of edge is 14 cm long has a circle on each faces painted. What is the total area remain unpainted of the cube, if the circle is the largest possible made on the face?
  1. 192 cm$^2$
  2. 222 cm$^2$
  3. 212 cm$^2$
  4. 242 cm$^2$
  5. 252 cm$^2$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Each face of the cube has area 14×14 = 196 cm². The largest circle on a face has diameter 14 cm, radius 7 cm, area π×7² = 154 cm². Unpainted area per face = 196 - 154 = 42 cm². Total unpainted area on 6 faces = 6 × 42 = 252 cm².

Multiple choice

The curved surface area and the total surface area of a cylinder are in the ratio 1 : 2. If the total surface area of the right cylinder is 616 cm.2, then its volume is एक बेलन के वक्र पृष्ठ क्षेत्रफल और कुल पृष्ठ क्षेत्रफल में अनुपात 1 : 2 है | यदि लम्बवृत्तीय बेलन का कुल पृष्ठीय क्षेत्रफल 616 सेमी.2, तो इसका आयतन है

  1. 1232 cm.3/सेमी.3

  2. 1848 cm.3/सेमी.3

  3. 1632 cm.3/सेमी.3

  4. 1078 cm.3/सेमी.3
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a cylinder: CSA = 2πrh, TSA = 2πr(r+h). Given CSA:TSA = 1:2, so 2πrh : 2πr(r+h) = 1:2, giving h : (r+h) = 1:2, thus r = h. With TSA = 616 = 2πr(r+r) = 4πr², we get r = 7. Volume = πr²h = π(7²)(7) = 1078 cm³. Option D matches.

Multiple choice

A cylindrical rod of iron whose height 8 times its radius is melted and cast into spherical balls each of half the radius of the cylinder. The number of such spherical balls is - एक बेलनाकार लोहे की छड़ जिसकी ऊँचाई त्रिज्या की 8 गुना है, को पिघलाया गया एवं गलाकर गोलीय गेंदे बनाई गयी जिनकी त्रिज्या बेलन की त्रिज्या की आधी है। ऐसी गोलीय गेंदों की संख्या है-

  1. 16

  2. 32

  3. 24

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

Let cylinder radius = r, so height = 8r. Volume of cylinder = πr²(8r) = 8πr³. Each sphere has radius r/2, so volume = (4/3)π(r/2)³ = (4/3)π(r³/8) = πr³/6. Number of spheres = (8πr³)/(πr³/6) = 8 × 6 = 48.

Multiple choice

What are the dimensions (length, breadth and height respectively) of a cuboid with volume 720 cubic cm, surface area 484 square cm and the area of the base 72 square cm?

  1. 9, 8 and 10 cm.

  2. 12, 6 and 10 cm.

  3. 18, 4 and 10 cm.

  4. 30, 2 and 12 cm.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given volume = l × b × h = 720, surface area = 2(lb + bh + lh) = 484, and base area = lb = 72. From surface area: 2(72 + bh + lh) = 484, so bh + lh = 170. Using volume: h = 720/72 = 10. Then b(l + h) doesn't help directly. Better: from bh + lh = 170 and h = 10, we get b + l = 17. With l × b = 72 and l + b = 17, the dimensions are 9 and 8. Testing option A: 9 × 8 × 10 gives volume 720 and surface area 2(72 + 80 + 90) = 484. ✓

Multiple choice

Curved surface area of a cylinder is 1232 sq cm. If circumference of its base is 154 cm, then what will be the height of the cylinder? (Take π= 22/7) एक बेलन के वक्र सतह का क्षेत्रफल 1232 वर्ग सेमी. है। यदि इसके आधार की परिधि 154 सेमी. है, तो उस बेलन की ऊँचाई कितनी होगी? (π= 22/7 लीजिए)

  1. 16 cm /सेमी.

  2. 4 cm /सेमी.

  3. 8 cm /सेमी.

  4. 12 cm /सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Curved surface area of cylinder = 2πrh = 1232. Circumference 2πr = 154, so r = 154/(2π) = 24.5. Substituting: 2π(24.5)h = 1232 gives h = 8 cm. Option A is 16 cm (double the correct answer), B is too small, and D is 12 cm.