Important Formulas for CAT Mensuration

Areas

Areas of some well-known figures are given below:

Example.1 A horizontal pipe for carrying floodwater has a diameter of 1 m. When water in it is 10 cm deep then the width of water surface AB is equal to ________ cm.

(1) 10 cm 
(2) 25 cm
(3) 60 cm 
(4) 70 cm
(5) None

Correct Answer is Option (3).

• Let O be the centre of circle.
• In ΔOAD
  ⇒ 502 = 402 + x²
  ⇒ 2500 – 1600 = x²
  ⇒ x² = 900
  ⇒ x = 30
• ∴ AB = 2x = 60 cm. 

Question for Important Formulas: Geometry

Try yourself:The area of quadrilateral PQRS is 228cm². Find the length of ?.

• A.

  16 cm
• B.

  9 cm
• C.

  8 cm
• D.

   18 cm

Explanation

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Example.2 Area of the shaded region is:

 Correct Answer is Option (1).

Volume

Length of a Diagonal

• The length of diagonal of a cuboid =  
• The length of a diagonal of a cube =

Example.3 An insect is crawling to climb a 20 metre long pipe of diameter 3.5 metres. If the insect take 4 circular rounds to reach the top of the building, then what will be total distance (in nearer metres) traveled by insect to reach the top?
(1) 47 m 
(2) 48 m 
(3) 49 m 
(4) 50 m

• Since the insect is moving in circular way, so it will cover the distance which will be 4 times the distance of the longest side AB of a right angled triangle. In the figure AO is equal to the circumference of the pipe.
  AO = πd =   = 11m
  BO = 5 m
  AB =  =  = 12.08 m
• Total distance traveled = 12.08 × 4 = 48.32 m
• Hence distance in nearer meters = 48 m
  

Example.4 The surface area of a cube is 726 sq metres. Find the volume of the cube
(1) 1234 m² 
(2) 1431 m²
(3) 1341 m³ 
(4) 1331 m³ 
(5) None of these

Correct Answer is Option(4).

• Surface area S = 726 m² Volume V = ?
• In a cube,
• S = 6a² = 726 ⇒ a² = 121 ⇒ a = 11 m
• So, V = a³ = 11³ = 1331 m³.