CAT Quantitative Aptitude topic i.e. Mensuration is the sub-topic of Geometry that holds around 20% weightage in CAT Question Paper. Questions based on this topic could be tricky but can be easily solved with sound knowledge of important formulas. Check CAT 2021 Syllabus

CAT 2021 spring candidates are advised to daily revise the formulas of Cube, Cuboid, Sphere etc, and are also advised to move to the next question if you had any confusion in the formulas during the time of paper. Some important formulas with 3D and 2D figures are provided in the article below. Candidates can also check the solved CAT sample questions related to the topic only to prepare for CAT 2021. 

Quick Links:

---

CAT Mensuration: Solids & their Important Formulas

• A solid has 3 sides - Length, Breadth/Width and Height. 
• A polyhedron is the figure that forms when the plane surfaces are bound. 
• LSA – Lateral Surface Area Lateral surface area
• TSA – Total Surface Area Total surface area

CAT Mensuration: Cuboids, Cube, and Cylinder 

Cuboids

• A cuboid has a length (l), breadth (b), and height (h).
• Formulas for Cuboids: 
  1. Volume = Area of base×height = lbh
  2. Volume = xyz
  3. Volume = xh = yl = zb
  4. Lateral surface area (LSA) or area of the four walls = 2 (l+b) h
  5. Total surface area (TSA) = 2(x+y+z) = 2(lb+bh+lh)
  6. Diagonal = l2 + b2 + h2

Cube

• A cube is a solid figure with 6 faces. 
• All faces form a square. 
• Length, Breadth and Height of a Cube is equal. 
• Formulas for Cube: 
  1. Volume = Cube of a
  2. Lateral surface area (LSA) or area of the four walls = square of 4a
  3. Total surface area (TSA) = square of 6a
  4. Diagonal = square of a

Right Circular Cylinder

• r is the radius of the base
• h is the height of right circular cylinder
• Formulas: 
  1. Volume = area of base×height
  2. Volume = r2h
  3. Curved surface area (CSA) = Perimeter of base×height
  4. LSA = 2p2rh
  5. Total surface area (TSA) = LSA + area of the top+area of the base
  6. TSA = 2rh+r2+r2
  7. TSA = 2r(r+h)

Quick Links:

---

CAT Mensuration Prism, Cones & Pyramids

• A Prism is solid with identical and parallel top and bottom faces. 
• The side faces of a prism are rectangular and are known as lateral faces. 
• The distance between two bases is known as the height or the length of the prism.

• Formula: 
  1. Volume = Area of base×Height
  2. Lateral surface area (LSA) = Perimeter of the base×Height
  3. Total surface area (TSA) = LSA+(2 ×Area of the base)

Right triangular prism

• Volume: Area of the base×Height 
• LSA: Perimeter of the base×Height 
• TSA: Lateral surface area+2(Area of base)

Right Circular Cone

• R is the radius of the base
• H is the height
• L is the slant height of the circular cone
• Formula: 
  1. Volume: ⅓ x Area of base height x height
  2. Volume: ⅓ r2h
  3. Curved Surface Area: rl
  4. Total Surface Area: CSA + Area of Base = rl + r2

Frustum of Cone

• TSA: Lateral surface area+Area of top+Area of base

Right pyramid

• Volume: 1/3 area of the base×Height
• CSA: 1/2×Perimeter of the base×Slant height
• TSA: Lateral surface area+Area of base

Quick Links:

• How to Prepare for CAT Seating Arrangement?
• How to Prepare for CAT Logical Reasoning?
• How to Prepare for CAT Venn Diagrams?

---

CAT Mensuration Spherical Shell & Spheres

Spherical Shell

• r = inner radius
• R = outer radius
• Volume: 4/3(R3-r3)
• TSA: 4(R2-r2)

Hemisphere

• r is the radius 
• Volume: ⅔ r2
• CSA: 2r2
• TSA: 3r2

Sphere

• r is the radius 
• Volume: 4/3 r2
• TSA: 4r2

Quick Links:

---

Mensuration Sample Questions for CAT

Ques: If there is a sphere that has the radius r and is cut by a plane at a distance of h from its center and the cut is that it has broken the sphere into two different pieces. What will be the value of h if the surface area of these two pieces is 25% more than that of the sphere?

1. r/√2
2. r/√3
3. r/√5
4. r/√6

Solution: 1

Ques: Two mutually perpendicular chords are named as AB & CD. These 2 chords meet at a point P. The point P is inside the circle such that AP = 6 cms, PB = 4 units, and DP = 3 units then calculate the area of the circle?

1. 125π/4 sq cms
2. 100π/7 sq cms
3. 125π/8 sq cms
4. 52π/3 sq cms

Solution: 1

Ques: You have to calculate the maximum number of cans that can fit into a box if we have to pack the cylindrical cans of cricket balls in a box. Let us consider that the radius of each can is 7 cm and height of 30 cm. Dimension of the box is l = 76 cm, b = 46 cm, h = 45 cm

1. 15
2. 17
3. 22
4. 21

Solution: 4

Ques: Let us wound a string around two circular disks as shown in the figure below. Consider the radius of the 2 disks is 40 cm and 30 cm respectively and then calculate the total length of the string?

1. 70 cm
2. 70 + 165*π cm
3. 70 + 120π cm
4. 70 + 165*π/2 cm

Solution: 4

Ques: Let us consider that the base of a pyramid is a square. Also, consider that the other four sides are equilateral triangles with the length of each side being 20 cm. Then what will be the vertical height of the pyramid?

1. 10√2
2. 8√3
3. 12
4. 5√5

Solution: 4

CAT Sample Questions for Preparation

Ques. If the side of a cube is increased by 100%, find by what percentage the surface area of the cube is increased?

(a) 150%

(b) 200%

(c) 300%

(d) 350%

Ques. What is the radius of a spherical ball in inches which is formed by melting a cylinder of base diameter 8 inches and height 160 inches, if the conversion wastage results in a 10% weight loss?

(a) 6

(b) 8

(c) 12

(d) 16

Ques. If the larger side of the tablecloth and the smaller side of the table are parallel, then what fraction of the area of the top of the table remains uncovered by the cloth?

(a) 3/7

(b) 4/9

(c) 7/11

(d) None of these

Ques. Three equal cubes of unit side length are placed adjacent to each other in a row. Find the ratio of the total surface area of the new cuboid to that of the sum of the surface areas of all the three cubes.

(a) 3:5

(b) 4:5

(c) 6:7

(d) 7:9

Ques. A cube of side length 3cm weighs 12kg. What is the weight of the similar cube of the same material whose side length is 12cm?

(a) 768kg

(b) 678kg

(c) 964kg

(d) 864kg

Ques. Inside a triangular garden, there is a flower-bed in the form of a similar triangle. Around the flower-bed runs a uniform path of such a width that the sides of the garden are double of the corresponding sides of the flower-bed. The areas of the path and the flower-bed are in the ratio

(a) 1:1

(b) 4:1

(c) 1:3

(d) 3:1

Ques. A rectangular piece of cardboard 18cm× 24 cm is made into an open box by cutting a square of 5cm side from each corner and building up the side. What is the volume of the box (in cm3)?

(a) 560

(b) 432

(c) 216

(d) None of these

Quick Links: