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

In a triangle ABC, AB=36 cm, AC=48 cm and the altitude to side BC measure 24 cm. Which of the following gives the area of the circle circumscribing the triangle ABC (In sq. cm)?

  1. 1296π

  2. 1176π

  3. 900π

  4. 576π
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of triangle = (1/2) * base * height. We have sides 36, 48 and altitude 24. Area = (1/2) * BC * 24 = 12 * BC. Also Area = (abc) / (4R). This requires finding BC. Using the altitude, we can find the segments of BC. The circumradius R = (abc) / (4 * Area).

Multiple choice

Find the volume, in cubic cm, of the largest possible cylinder that can be carved out of a sphere whose radius is 2√3 cm.

  1. 32π

  2. 16π

  3. 48π

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a cylinder of radius r and height h in a sphere of radius R, r^2 + (h/2)^2 = R^2. Volume V = pi * r^2 * h = pi * (R^2 - h^2/4) * h = pi * (12h - h^3/4). Maximize: dV/dh = pi * (12 - 3h^2/4) = 0 => h^2 = 16 => h = 4. r^2 = 12 - 4 = 8. V = pi * 8 * 4 = 32 * pi.

Multiple choice

A cube is cut into 64 identical smaller cubes. On one face of each smaller cube, 25 identical circles of maximum radius are drawn in a square grid. The sum of the diameters of four circles is 20 units. Find one-fourth of the maximum volume of the largest cylinder that can fit inside the original cube.

  1. 7,000π

  2. 5,000π

  3. 62,500π

  4. 15,625π

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

The problem is poorly defined. If 25 circles of max radius are in a square grid on a face, the side of the square is 10*diameter. The sum of 4 diameters is 20, so diameter = 5. Side of small cube = 25. Original cube side = 4 * 25 = 100. Largest cylinder in cube has diameter 100 and height 100. Volume = pi * r^2 * h = pi * 50^2 * 100 = 250,000 * pi. One-fourth of this is 62,500 * pi.

Multiple choice

A craftsman has blocks of three different metals: cobalt (816 cm3), nickel (1152 cm3), and zinc (624 cm3). He melts down each metal separately to cast solid closed conical objects (with base), all having the same height and a base radius of 4 cm. If the craftsman wants to minimise the total number of cones while using all the material, what is the combined surface area of all the cones (including the base), in square centimetres?

  1. 846π + 216 16 π 2 − 18

  2. 486π + 126 81 π 2 − 16

  3. 846π + 126 81 π 2 − 16

  4. 864π + 216 16 π 2 + 81

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

The greatest common usable cone volume is gcd(816, 1152, 624) = 48 cm^3. This gives 54 cones, height 9/pi cm, and total surface area 864pi + 216sqrt(16pi^2 + 81) cm^2.

Multiple choice

An engineer needs to design a rectangular display board that must fit entirely within a semi-circular frame of radius 5 cm, maximising the area of the board. What will be the ratio of the board's shorter side to its longer side when its area is the largest possible?

  1. 1 : 3

  2. 2 : 5

  3. 1 : 2

  4. 2 : 3

  5. 3 : 5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the rectangle have width 2x and height y. It is inscribed in a semicircle of radius R=5. By Pythagorean theorem, x^2 + y^2 = R^2 = 25. Area A = 2xy = 2x * sqrt(25 - x^2). To maximize, dA/dx = 0. A^2 = 4x^2(25 - x^2) = 100x^2 - 4x^4. Derivative = 200x - 16x^3 = 0 => x^2 = 200/16 = 12.5. Then y^2 = 25 - 12.5 = 12.5. So x = y. The sides are 2x and y = x. Ratio of shorter side (x) to longer side (2x) is 1:2.

Multiple choice

A regular right square pyramid has a square base with side length 16 cm and a slant height of 10 cm. What is the volume of the pyramid?

  1. 384 cm3

  2. 448 cm3

  3. 512 cm3

  4. 576 cm3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The height h of the pyramid is found using the slant height s=10 and half the base side a/2=8. By Pythagorean theorem, h = sqrt(10^2 - 8^2) = sqrt(100 - 64) = sqrt(36) = 6 cm. The volume is (1/3) * base_area * height = (1/3) * (16 * 16) * 6 = 256 * 2 = 512 cm^3.

Multiple choice

A right triangular prism has a right triangular base with legs measuring 3w centimeters and 4w centimeters. The area of the base is 6w2 square centimeters (cm2). The height of the prism is 10 cm. Which expression represents the surface area (in cm2) of the right triangular prism?

  1. 6w2 + 120w

  2. 12w2 + 210w

  3. 8w2 + 70w

  4. 12w2 + 120w
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Base area = 6w^2. Perimeter = 3w + 4w + 5w = 12w. Lateral area = Perimeter * height = 12w * 10 = 120w. Surface area = 2 * Base area + Lateral area = 2(6w^2) + 120w = 12w^2 + 120w.