Mensuration
Mensuration is the branch of mathematics that studies the measurement of the 2D and 3D figures on parameters like length, volume, shape, surface area, etc. In other words, it is the process of measurement based on algebraic equations and mathematical formulas. Let us learn more about the concept of mensuration, the formulas and solve a few examples to understand it better.
1.  What is Mensuration? 
2.  Uses of Mensuration 
3.  Important Mensuration Terms 
4.  Mensuration Formulas 
5.  FAQs on Mensuration 
What is Mensuration?
Mensuration can be explained as an act of measurement. We live in a threedimensional world. The concept of measurement plays an important role in primary as well as secondary school mathematics. Moreover, measurement has a direct connection to our everyday lives. When learning to measure objects we learn to do so for both 3D shapes and 2D shapes. Objects or quantities can be measured using both standard and nonstandard units of measurement. For example, a nonstandard unit of measuring length would be handspans. You can even do an activity on it by asking children to measure the length of objects using handspans. Let children notice that while measuring objects using nonstandard units there will always be a scope of a discrepancy. Hence the need for standard units of measurement. To measure parameters like length, weight, and capacity we now have units like kilometer, meter, kilogram, gram, liter, milliliter, etc.
3D Shapes Definition
A shape or a solid that has three dimensions is called a 3D shape that has faces, edges, and vertices. They have a surface area that includes the area of all their faces. The space occupied by these shapes gives their volume. Some examples of 3D shapes are cube, cuboid, cone, cylinder and some realworld examples are a book, a birthday hat, and, a coke tin.
2D Shapes Definition
In geometry, 2D shapes can be defined as plane figures that are completely flat and have only two dimensions – length and width. They do not have any thickness and can be measured only by the two dimensions.
Uses of Mensuration
Mensuration is an important topic with high applicability in reallife scenarios. Given below are some of the scenarios.
 Measurement of agricultural fields, floor areas required for purchase/selling transactions.
 Measurement of volumes required for packaging milk, liquids, solid edible food items.
 Measurements of surface areas required for estimation of painting houses, buildings, etc.
 Volumes and heights are useful in knowing water levels and amounts in rivers or lakes.
 Optimum cost packaging sachets for milk etc. like tetra packing.
Important Mensuration Terms
Mensuration deals with the measurement of plane shapes and solid shapes. Let us see some of the important terms used:
Terms  Definition 

Area  Area is the amount of space occupied by a twodimensional figure. It is expressed in square units. 
Perimeter  Perimeter is the total distance around the shape or the length of the boundary of any closed shape. It is expressed in square units. 
Volume  Volume is the amount of space occupied by a 3D shape. It is expressed in cubic meter. 
Surface Area  Surface Area is the total area occupied by the surfaces of a 3D object. They are classified into two  Curved or Lateral Surface Area and Total Surface Area. 
Mensuration Formulas
Mensuration formulas involve both 3D and 2D shapes. The most commonly used formula is the surface area and volume of these shapes. However, let us learn all the formulas for these shapes.
3D Shape Formulas
The following table shows different 3D shapes and their formulas.
3D Shape  Formulas 

Sphere  Diameter = 2 × r; (where 'r' is the radius) Surface Area = 4πr^{2} Volume = (4/3)πr^{3 } 
Cylinder  Total Surface Area = 2πr(h+r); (where 'r' is the radius and 'h' is the height of the cylinder) Volume = πr^{2}h 
Cone  Curved Surface Area = πrl; (where 'l' is the slant height and l = √(h^{2} + r^{2})) Total Surface Area = πr(l + r) Volume = (1/3)πr^{2}h 
Cube  Lateral Surface Area = 4a^{2}; (where 'a' is the side length of the cube) Total Surface Area = 6a^{2} Volume = a^{3} 
Cuboid  Lateral Surface Area = 2h(l + w); (where 'h' is the height, 'l' is the length and 'w' is the width) Total Surface Area = 2 (lw + wh + lh) Volume = (l × w × h) 
Prism  Surface Area = [(2 × Base Area) + (Perimeter × Height)] Volume = (Base Area × Height) 
Pyramid  Surface Area = Base Area + (1/2 × Perimeter × Slant Height) Volume = [(1/3) × Base Area × Altitude] 
2D Shape Formulas
The following table shows the formulas that are used to calculate the area and perimeter of a few common 2D shapes:
2D Shape  Area Formula  Perimeter Formula 

Circle  A = π × r^{2}, where 'r' is the radius of the circle and 'π' is a constant whose value is taken as 22/7 or 3.14  Circumference (Perimeter) = 2πr 
Triangle  Area = ½ (Base × height)  Perimeter = Sum of the three sides 
Square  Area = Side^{2}  Perimeter = 4 × side 
Rectangle  Area = Length × Width  Perimeter = 2 (Length + Width) 
Important Notes:
 Mensuration and measurement are learned together when an object is measured using nonstandard unit of measurement.
 Solid shapes and nets help in measuring an object. Solid shapes will help understand faces, edges, and vertices. Nets will help in visualizing the structure of 3D shapes.
ā Related Topics:
Mensuration Examples

Example 1: Find the area of a square with a side of 5 cm.
Solution:
Area of a square = side × side. Here, side = 5 cm
Substituting the values, 5 × 5= 25.
Therefore, the area of the square = 25 square cm.

Example 2: Find the surface area of a cuboid of length 4 units, width 5 units, and height 6 units.
Solution:
Given that, length of the cuboid = 4 units, width of the cuboid = 5 units, height of the cuboid = 6 units
Surface area of the cuboid is 2 × (lw + wh + lh) square units
= 2 × (lw + wh + lh)
= 2[(4 × 5) + (5 × 6) + (4 × 6)]
= 2(20 + 30 + 24)
= 2(74)
= 148 square units.
Therefore, the surface area of the cuboid is 148 square units.

Example 3: Find the area of a circle whose radius is 6 cm.
Solution: Yes, a circle comes under the category of 2D shapes. The area of a circle = π × r^{2}; where 'r' is the radius of the circle and π is a constant whose value is 22/7 or 3.14.
Area of the circle = π × r^{2}
= 3.14 × 6^{2}
= 3.14 × 36
Therefore, the Area of the circle = 113.04 square cm.
FAQs on Mensuration
Who Introduced Mensuration?
Archimedes is remembered as the greatest mathematician of the ancient era. He contributed significantly in geometry regarding the area of plane figures and areas as well as volumes of curved surfaces.
What is Mensuration in Math?
Mensuration in maths deals with the geometric qualities of 2D and 3D shapes such as the area, volume, and perimeter. In other words, the study of measurements of these shapes is known as mensuration.
What is the Difference Between Mensuration and Geometry?
Mensuration refers to the calculation of various parameters of shapes like the perimeter, area, volume, etc. Whereas, geometry deals with the study of properties and relations of points and lines of various shapes.
What are 2D and 3D Mensuration?
2D mensuration is the calculation of 2D shapes on different parameters such as area and perimeter. Whereas 3D mensuration is the study of volume and lateral surface area of 3D shapes.
What is Mensuration Formula?
Mensuration formulas involve the formulas used to calculate the different parameters of 2D and 3D shapes such as the area, perimeter, and volume. You can read about the detailed list of formulas of both these shapes in the previous section of this article.
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