# CAT Questions | Geometry CAT Questions: Mensuration

## Importance of Mensuration

Mensuration in Geometry is a fairly intuitive subject. Structural and spatial understanding of the forms, other than knowing simple formulae, makes one nail Mensuration questions in the test. A candidate can easily expect 3-5 Mensuration questions in CAT based on logarithm and also in other MBA exams such as NMAT, XAT, SNAP, etc.

## CAT Quantitative Aptitude questions | CAT Geometry: Mensuration

1. A solid right circular cone of height 27 cm is cut into pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

A) 243        B) 232         C) 256      D) 264                              [CAT 2020]

If height becomes 1/3rd then radius also becomes 1/3rd

If both height and radius become 1313rd, Volume will be 1/27th

So, Top part’s volume = V

Remaining part = 26V

Totally = 27V

Given 26 V – V = 225

V = 9

27 V = 243

2. The length, breadth, and height of a room are in the ratio 3:2:1. If the breadth and height are halved while the length is doubled, then the total area of the four walls of the room will (CAT 19 – Slot 1)

1. Remain the same

2. Dec. by 13.64%

3. Dec. by 15%

4. Dec. by 18.75%

5. Dec. by 30%

in the present case, let length = l = 3x, breadth = b = 2x, height = h = x

then, area of four walls = 2 (l + b) h = 2(3x + 2x) x = 10×2.

now as length gets doubled = 6x, breadth halved = x, height halved = x/2.

new area of four walls = 2 (6x + x) x/2 = 7×2.

thus there is a decrease of 30%. hence, the fifth option is the answer.

3. Corners are cut off from an equilateral triangle T to produce a regular hexagon H. Then, the ratio of the area of H to the area of T is

A) 3:4        B) 2:3         C) 5:6      D) 4:5                              [CAT 2019]

In given case , figure can be drawn as below

Let side of equilateral triangle ABC = 3a

So side of hexagon = a

Area of triangle = (root3 /4 )*(3a)^2 = 9a^2 *(root3/4)

Area of hexagon with side a = 6*(root3 /4 )*(a)^2

Ratio of areas of hexagon to that of triangle = { 6*(root3 /4 )*(a)^2 } / { 9a^2 *(root3/4)} = 6/9 = 2/3

Thus required ratio = 2 : 3

4. A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is                               [CAT 2019]

A) 8464π          B) 928π         C) 1026(1 + π)       D) 1044(4 + π)

As cylinders have the same volume and each of these has radius 3 cm. So volume of each cylinder will be equal to HCF of (405, 783 and 351) which is 27.

So volume of each cylinder = 27

No of cylinders = [ 405/27 ] + [783/27] + [351/27] = 15 + 29 + 13 =57

Using V = πr^2 h

27 = 22/7 *9 *h

So h = 3/ π cm

## List of concepts

• Impact of the cavity on a solid figure’s surface area

• Length of longest rod in cube shape room

• Impact of the solids melting to create new solids

• Slant surface and Entire Pyramid surface

• Dimensions of Pyramid etc.

### Important Formulas

Learn, study and remember the Curved Surface Area, Gross Surface Area, and Volume for the following solid figures or mensuration topics

• Cube
• Cuboid
• Cylinder
• Cone
• Sphere
• Frustum
• Polygon
• Prism
• Pyramid etc
 SOLID Total Surface Area Lateral/ Curved Surface area Volume Length of Leading Diagonal/ Slant Height Cube 2(LB+ BH+ HL) 2H (L + B) LBH √ (L2 + H2 + B2) Cuboid 6a2 4a2 a3 √3a Cylinder 2Πr (r + h) 2Πrh Πr2h No Slant height or diagonal Cone Πr (r + l) Πrl ⅓Πr2h √(h2 + r2) Sphere 4Πr2 4Πr2 4/3Πr3 No Slant height or diagonal Hollow Cylinder 2Π(r₁+r₂) (r₂-r₁+h) 2Πh(r₁+r₂) Πh(r₂²-r₁²) No Slant height or diagonal Frustum Π(R1 + R2)s + (R12 + R22) Π(R1 + R2)s ⅓Πh(R12 + R22 + R1R2) √(h2 + (R1 – R2)2) Hemisphere 3Πr2 2Πr2 2/3Πr3 No Slant height or diagonal

## How to prepare this topic for CAT?

Level 1

• •Use Formula and Mensuration shortcuts. Understand how geometry word issues work. A diagram of what you understand from a specific query is the best way to do this. Let yourself learn the basic concepts and tricks in geometry from MBAP CAT E Book (Concept Theory) study material. For revision, you can use MBAP Live Lecture Recording (Basic) on basic concept and MBAP lecture Assignment.
• Mensuration problems can be answered several times with the use of SYMMETRY in measurement. CAT geometry Questions involving solid figures such as cubes, circles, pyramids, etc. can be solved in a simpler way using the principle of SYMMETRY, but good practice and strong analysis is required for that.

Level 2

Write down the formulas on a sheet of paper and put them on the wall so you can see them regularly. It’ll help memorize them for you.

• To understand diagrams well, you must also practice multiple problems on mensuration. You can choose MBAP CAT E Book (Practice Questions). Alternatively, Arun Sharma’s ‘How to Prepare for Quantitative Aptitude for the CAT’ is a well known for CAT Geometry training. It is elaborate and easy to understand the examples given in it. To help you organize your training accordingly, it is also split into three stages of difficulty.

Level 3

• Including all the above levels make sure you go through all the different Mensuration questions from the previous years of different exams, since so many questions are based on the concepts that have appeared in previous entrance exams. For this you can use MBAP Topic wise Previous Year CAT Question and MBAP Previous Year CAT paper. Spend lots of time per day to practice Geometry for CAT and practice mensuration problems for competitive exams other than CAT you intend to appear in particular. The more time you spend on the questions, the more familiar they’ll get.

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