Important Formulas: 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

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

Question for Important Formulas: Mensuration
Try yourself:The area of quadrilateral PQRS is 228cm2. Find the length of QA.

Example.2 Area of the shaded region is:

## 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

Question for Important Formulas: Mensuration
Try yourself:What is the volume of a cone having radius of 21cm and height of 5cm?

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

• Surface area S = 726 m2 Volume V = ?
• In a cube,
• S = 6a2 = 726 ⇒ a2 = 121 ⇒ a = 11 m
• So, V = a3 = 113 = 1331 m3.
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## FAQs on Important Formulas: Mensuration - CSAT Preparation - UPSC

 1. What is mensuration?
Ans. Mensuration is a branch of mathematics that deals with the measurement of geometric figures such as areas, volumes, and lengths.
 2. What are the important formulas used in mensuration?
Ans. Some important formulas used in mensuration are: - Area of a rectangle: Length × Width - Area of a triangle: 1/2 × Base × Height - Area of a circle: π × Radius^2 - Volume of a cylinder: π × Radius^2 × Height - Volume of a sphere: 4/3 × π × Radius^3
 3. How do you calculate the area of a triangle?
Ans. To calculate the area of a triangle, you can use the formula: 1/2 × Base × Height. The base is the length of the triangle's bottom side, and the height is the perpendicular distance from the base to the opposite vertex.
 4. What is the formula for finding the volume of a cylinder?
Ans. The formula for finding the volume of a cylinder is: π × Radius^2 × Height. The radius is the distance from the center of the circular base to its edge, and the height is the distance between the two circular bases.
 5. How can I calculate the volume of a sphere?
Ans. To calculate the volume of a sphere, you can use the formula: 4/3 × π × Radius^3. The radius is the distance from the center of the sphere to any point on its surface.

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