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Mensuration is a vital branch of mathematics that delves into the art of measuring geometric figures and understanding their various parameters. From calculating lengths and volumes to exploring surface areas and shapes, mensuration equips us with the tools to comprehend the intricate characteristics of diverse geometric objects.
This article will unravel the essential concepts of mensuration, mensuration formulas and properties for different geometric shapes.
Mensuration Maths – Definition of Mensuration
Mensuration in mathematics is a specialised branch that focuses on the measurement and study of geometric shapes’ length, volume, and area. These shapes can exist in either two dimensions (2D) or three dimensions (3D). In 2D, mensuration deals with figures such as triangles, circles, squares, and rectangles, while in 3D, it encompasses objects like cubes, spheres, cylinders, and prisms.
Mensuration: Differences Between 2D and 3D Shapes
Below mentioned table marks the differences between 2D and 3D shapes with respect to different aspects.
| Aspect | 2D Shapes | 3D Shapes |
| Dimension | Exist in two dimensions (length and width) | Exist in three dimensions (length, width, and height) |
| Examples | Circle, Square, Triangle, Rectangle, etc. | Cube, Sphere, Cylinder, Prism, etc. |
| Representation | Flat and drawn on a plane | Solid and occupy space |
| Properties | Area is the main property | Volume and surface area are the main properties |
| Measurements | Length and area | Length, area, and volume |
| Visual representation | Appear as flat shapes on paper | Appear as solid objects in space |
| Real-world applications | Measuring floor area, drawing shapes, calculating perimeters, etc. | Measuring volumes, designing buildings, creating 3D models, etc. |
Let’s have a look at the terms, their abbreviations and definition which will be related to mensuration.
| Terms | Abbreviation | Unit | Definition |
| Area | A | m² or cm² | The surface covered by a closed shape. |
| Perimeter | P | cm or m | Measures the continuous line along the boundary of a figure. |
| Volume | V | cm³ or m³ | The space occupied by a 3D shape. |
| Curved Surface Area | CSA | m² or cm² | The total area of a curved surface in a 3D shape (e.g., Sphere). |
| Lateral Surface Area | LSA | m² or cm² | The total area of all lateral surfaces surrounding a 3D figure. |
| Total Surface Area | TSA | m² or cm² | The area covered by a square of one unit side length. |
| Square Unit | – | m² or cm² | The area covered by a square of one unit side length. |
| Cube Unit | – | m² or cm² | The volume occupied by a cube with one side of one unit length. |
Mensuration Formulas For 2D Shapes
Below are the mensuration formulas for two-dimensional shapes in geometry.
| Shape | Area (Square Units) | Perimeter (Units) |
| Square | a² | 4a |
| Rectangle | l × b | 2(l + b) |
| Circle | πr² | 2πr |
| Scalene Triangle | √[s(s−a)(s−b)(s−c)], Where, s = (a+b+c)/2 |
a + b + c |
| Isosceles Triangle | (√3/4) × a² | 3a |
| Right Angle Triangle | ½ × b × h | b + hypotenuse + h |
| Rhombus | ½ × d₁ × d₂ | 4 × side |
| Parallelogram | b × h | 2(l + b) |
| Trapezium | ½ × h(a + c) | a + b + c + d |
Mensuration Formulas for 3D Shapes
Go through the mensuration formulas tabulated below for three-dimensional shapes in geometry.
| Shape | Volume (Cubic Units) | Curved Surface Area (CSA) or Lateral Surface Area (LSA) (Square Units) | Total Surface Area (TSA) (Square Units) |
| Cube | a³ | CSA = 4a² | 6a² |
| Cuboid | l × b × h | LSA = 2h(l + b) | 2(lb + bh + h |
| Sphere | (4/3)πr³ | CSA = 4πr² | 4πr² |
| Hemisphere | Hemisphere (⅔)πr³ |
LSA = 2πr² | 3πr² |
| Cylinder | πr²h | LSA = 2πrh | 2πrh + 2πr² |
| Cone | (⅓)πr²h | πrl | πr(r + l) |
Mensuration Solved Problems
Question 1: Find the area and perimeter of a rectangle with length 10 cm and width 6 cm.
Solution: Given:
Length of the rectangle (l) = 10 cm
Width of the rectangle (b) = 6 cm
Area of the rectangle = length × width
Area = 10 cm × 6 cm
= 60 square centimetres (cm²)
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (10 cm + 6 cm)
= 32 centimetres (cm)
Therefore, the area of the rectangle is 60 cm², and the perimeter is 32 cm.
Question 2: A cylindrical water tank has a height of 15 meters and a base radius of 4 meters. Calculate the tank’s volume and total surface area (take π = 3.14).
Solution: Given:
Height of the cylinder (h) = 15 meters
Base radius of the cylinder (r) = 4 meters
π (Pi) = 3.14 (approximate value)
Volume of the cylinder = πr²h cubic meters
Volume = 3.14 × 4² × 15 = 753.6 cubic meters (approx.)
Total Surface Area of the cylinder = 2πrh + 2πr² square meters
TSA = 2 × 3.14 × 4 × 15 + 2 × 3.14 × 4²
= 376.8 + 100.48
= 477.28 square meters (approx.)
Therefore, the volume of the cylinder is approximately 753.6 cubic meters, and the total surface area is approximately 477.28 square meters.
Frequently Asked Questions on Mensuration
What is mensuration, and example?
Mensuration is the branch of mathematics that deals with measuring geometric figures like length, area, volume, etc. Example: Calculating the area of a rectangular field.
What is the method of mensuration?
The mensuration method involves using specific formulas and equations to measure and calculate the dimensions and properties of geometric shapes based on given information.
What are the two types of mensuration?
The two types are: - 2D (Two-Dimensional) Mensuration: Deals with flat shapes like squares, circles. - 3D (Three-Dimensional) Mensuration: Involves solid shapes like cubes, spheres.
What is mensuration formulas?
Mensuration formulas are mathematical equations used to calculate parameters like area, volume, and perimeter of geometric shapes.
