Have you ever thought about how much parking space your car needs? Or what dimension bed can fit in your room? The answer to these questions lies in mensuration formulas. In Latin, mensuration means measurement. Hence, mensuration is that branch of mathematics that deals with the calculation of the length, area, and volume of both 2D and 3D geometric shapes. But why do you need to learn these formulas now? Quantitative reasoning section of exams like GRE, GMAT, SSC, RBI Grade B, etc. is considered as one of the toughest yet highly scoring section. Thus, speed and accuracy play a pivotal role in scoring well. Now here comes the role of memorizing mensuration formulas. If you can recall a formula, then you can invest more time on other questions. Here is a blog that compiles all the relevant mensuration formulas for competitive exams.
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What is Mensuration?
Mensuration is a prominent concept of Mathematics and studies the measurement of different geometric shapes and figures. As discussed, mensuration formulas come handy in scores of competitive exams. It helps us understand the dimensions of various 2-dimensional and 3-dimensional objects. While a 2D shape has only two dimensions i.e. length and breadth, a 3D figure has length, breadth, and height. Area(A) and Perimeter(P) are the two common parameters we measure for 2D shapes. For 3D, Volume(V), total, lateral and curved surface area is calculated.
Difference Between 2D & 3D Shapes
Before diving it mensuration formulas, it is important to understand the two major types of geometric shapes, i.e. Two-dimensional (2D) and three-dimensional (3D).
|2D Shapes||3D Shapes|
|A 2D shape refers to a figure which is surrounded by three or multiple straight lines in a plane.||A 3D shape is a figure covered by multiple surfaces or planes.|
|2D shapes do not contain any depth or height.||3D shapes are solid shapes and have depth as well as height.|
|They comprise 2D length and breadth.||They consist of length, breadth and width as they are three-dimension.|
|Area and perimeter of these shapes are measured.||Volume, CSA, LSA and TSA is measured for these shapes.|
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Mensuration Formulas: Important Terms
Before we get down to the nitty-gritty of the mensuration formulas, let us recall some important terms:
|Area (A)||It is the surface enclosed by a given shape.||m2 or cm2|
|Perimeter (P)||It is simply the boundary length of an area.||m or cm|
|Volume (V)||The space occupied by a solid or |
a 3-Dimensional object is called volume.
|cm3 or m3|
|Curved Surface Area (CSA)||It is the area enclosed by the curved portion |
of a geometrical object.
|m2 or cm2|
|Total Surface Area (TSA)||The sum total of areas of all the surfaces |
of an object is called TSA.
|Lateral Surface Area (LSA)||Sum total of areas of all surfaces except |
the top and the base of an object is called LSA.
|Diagonal (d)||A line that joins two vertices of a geometrical |
figure is called a diagonal.
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Mensuration Formulas for 2D Shapes
The major 2D figures are square, triangle, rectangle, circle, rhombus and parallelograms. Let us now have a look at the mensuration formulas of all the important 2D geometrical figures:
|a2||4a||√2a||Side = a|
|l x b||2(l+b)||√2 (l2+b2)||Length = lBreadth = b|
|½ × d1 × d2||4a||2A/d2||Diagonals = d1 and d2|
|p x h||2(p+q)||√(p2+q2-2pqcosβ)||Base = pSide = qAngle = β|
|–||Radius = r|
Types of Triangles
Triangles are another important 2D shapes whose formulas are important to memorize. But before we delve into the mensuration formulas of these objects, let us first understand their types.
- The triangle in which neither of the three sides and angles are of the same value is called a Scalene Triangle. In this, the sum total of all the angles equals 180 degrees.
- An Isosceles Triangle is the one in which any of the 2 sides are equal and the values of 2 angles are also equal.
- The triangle whose all sides are equal is called an Equilateral Triangle. Here, the value of all the 3 angles is 60 degrees.
- A triangle in which one angle equals 90 degrees is called a Right-angled Triangle. The side in this triangle is calculated using the Pythagoras theorem.
Now that you are familiar with different types of triangles, here are the mensuration formulas for Scalene, Right-Angled, Isosceles, and Equilateral triangles:
|Scalene||(bh)/2||a+b+c||Sides = a,b,c|
Height = h
|Isosceles||1/2 x b x h|
|2a+b||Semi-perimeter = S|
|Equilateral||a2 x √3/4||3a||Side = a|
*If sides are mentioned then the area of a scalene triangle is calculated using Heron’s formula where S=(a+b+c)/2
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3D Figures Mensuration Formulas
Below is a compilation of mensuration formulas of all the important 3D geometrical figures:
|Shapes||Volume||Curved Surface Area/Lateral Surface Area||Total Surface Area||Nomenclature|
|4/3 πr3||4 πr2||4πr2||Radius = r|
|a3||4 x a2||6a2||Side = a|
|l x b x h||2h(l+b)||2(lb+bh+hl)||Length = l|
Breadth = b
height = h
|πr2 x h||2πrh||2πr(r+h)||Radius of base = r|
|1/3πr2h||πrl||πr(s+l)||Slant height = s|
|4/6 πr3||3πr2||2πr2||Radius = r|
Mensuration Formulas: Solved Examples
Here are some important solved examples for you to help you understand the mensuration formulas better-
Q. Find the perimeter of a rectangular park with length 20 cm and breadth 40 cm.
Length of the rectangular park = 20 cm
Breadth of the rectangular park =40cm
Perimeter of a rectangle= 2 (L+B)
=2 (20 + 40) cm
= 2(60) cm
= 120 cm
The perimeter of the rectangular park is 120 cm
Q. Riya carries water to the school in a cylindrical flask with radius 4 cm and height 28 cm. Determine the amount of water she carries in the flask.
Radius of cylindrical flask= 4 cm
Height of cylindrical flask= 28 cm
volume of cylindrical flask = πr2 x h
=22/7 x 4x4x 21
= 22 x4x4x3
= 1,056 cubic cm
Q. Rohit stays in a cuboidal hotel room with dimensions 21x 10x 8. Find the total surface area of the room.
Length of the room= 21cm
Breadth of the room= 10 cm
Height of the room= 8cm
Total surface area of a cuboid= 2 (LB + BH + LH)
= 2 (21×10 + 10×8 + 21×8)
= 2 (210 + 80 + 168)
= 2 (458)
= 916 cubic cm
Practice Questions for 3D Mensuration
Now that you have understood the formulas, let us solve these questions-
- There is a water tank which is 300 metres long and 200 metres wide. water flows through this tank with the cross-section 0.3 x 0.5m 15 kilometres per hour. Determine the time in hours in which the water level will reach up to 10m.
- A cube of 10 cm has to be painted along all its sides. If this cube is sliced into multiple cubes of 1 cm, how many cubes will have exactly one of their faces painted?
- From a solid sphere of diameter d, a cube of the maximum possible volume is cut out. Then, what will be the volume of the remaining waste material of the square?
- All the 5 faces of a given regular pyramid having a square base have the same area. 5 cm is the height of the pyramid, calculate its total area of all the surfaces.
- What will be the dimensions of the side of a cube that can be inscribed into a cone having radius 6 and height 8cm?
- A hemisphere has been melted and reformed in the shape of a cone having the same base radius as 20 cm. If the height of the cone is H, find the relationship between the radius and height of the cone.
- A metallic spherical ball has been reconstructed into a cone. The diameter of bol was 6cm whereas the diameter of the base of the cone is 12cm. What is the height of the cone?
- The volume of a cylinder is 1500 cubic metres. The radius of its base is 50m. Determine what will be the height of the cylinder.
- Siddharth has decided to get his room painted. The dimensions of his walls are 20m x 40m x 12m. The painters have charged Rs. 20 per square metre. What will be the exact amount that Siddharth has to pay to the painters for painting his room?
- A cone of height 20 cm and diameter 40cm has been cut out of a wooden sphere having radius 25cm. Determine the percentage of wood that is wasted.
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Q1: How do you memorize mensuration formulas?
The best way to memorize mensuration formulas is by practising various questions on the same. The more questions you will solve, the easier it will be to retain a particular formula. Practising these questions regularly will also improve your calculation speed.
Q2: How many formulas are in mensuration?
There are 10 basic mensuration formulas in Mathematics out of which, 5 for 2D figures and 5 are for 3D figures.
Q3: What are the topics in mensuration?
Mensuration deals with obtaining perimeter, area and volume of different kinds of geometric figures. Both 2D and 3D figures are included in mensuration.
Q4: What is called mensuration?
Menstruation is a field of Mathematics which includes the study of various geometric figures. Both 3D and 2D objects are included in this. Based on this process of measurement, we are able to calculate the area, volume and perimeter of different shapes.
Thus, we have provided you with a list of all the relevant mensuration formulas for competitive examinations. If you want any guidance on how to score well in these exams, you can contact mentors and counselors at Leverage Edu.