Mensuration 2D & 3D Concepts, Formulas & Tricks for SSC by Ravinder Sir
Mensuration 2D & 3D concepts by Ravinder Sir details triangle classifications, area, perimeter, inradius, and circumradius for equilateral, isosceles, scalene, and right-angled types. Circle properties, sectors, and composite figures are discussed by Ravinder Sir, alongside cylinder and cone formulas and shortcuts for SSC exam.
Mensuration is one of the most scoring and practical topics in SSC examinations. From calculating areas of fields to finding volumes of tanks, the concepts of 2D and 3D mensuration are widely applied in real-life as well as competitive exams.
In “Mensuration 2D & 3D Concepts, Formulas & Tricks for SSC” by Ravinder Sir, students are guided through a complete, exam-focused approach to mastering this important subject.
Mensuration Introduction
Mensuration, the branch of mathematics dealing with geometric measurements, is fundamental for competitive exams. It helps us calculate area, perimeter, surface area, and volume of different shapes used in daily life, such as fields, rooms, tanks, roads, and buildings.
In competitive exams like SSC CGL, CHSL, MTS, and others, Mensuration is considered a high-scoring and concept-based topic. Questions are generally formula-based but require clarity, accuracy, and speed. With proper understanding and smart tricks, this topic becomes one of the easiest sections in Mathematics.
Triangles (त्रिभुज)
Triangles (त्रिभुज) are closed figures formed by three sides, three angles, and three vertices. Despite its simple structure, it contains powerful properties and theorems that frequently appear in exams.
From basic angle sum property (Sum of angles = 180°) to important concepts like congruency, similarity, Pythagoras theorem, and area formulas, triangles play a central role in problem-solving.
Classification of Triangles
Triangles are classified into different types based on the length of their sides and the measure of their angles.
1. Classification Based on Sides (भुजा के हिसाब से)
Equilateral Triangle (समबाहु त्रिभुज): A triangle where all three sides are of equal length.
Isosceles Triangle (समद्विबाहु त्रिभुज): A triangle where two sides are of equal length.
Scalene Triangle (विषमबाहु त्रिभुज): A triangle where all three sides have different lengths.
2. Classification Based on Angles (कोण के हिसाब से)
Acute Angle Triangle (न्यून कोण त्रिभुज): A triangle where all three angles are less than 90°.
Right-Angled Triangle (समकोण त्रिभुज): A triangle with one angle exactly equal to 90°.
Obtuse Angle Triangle (अधिक कोण त्रिभुज): A triangle with one angle greater than 90°.
Equilateral Triangle (समबाहु त्रिभुज)
An equilateral triangle has three equal sides (length 'a') and three equal angles (60° each).
Area (क्षेत्रफल): Area = (√3 / 4) * a²
Height (ऊंचाई): Height (h) = (√3 / 2) * a
Perimeter (परिमाप): Perimeter = 3a
Inradius and Circumradius of an Equilateral Triangle
Inradius (r): The radius of the circle inscribed inside the triangle. r = a / (2√3)
Circumradius (R): The radius of the circle that passes through all three vertices. R = a / √3
Important Relationships:
The ratio of the inradius to the circumradius (r : R) in any equilateral triangle is a fixed constant: r : R = 1 : 2.
The ratio of the area of the incircle to the area of the circumcircle is: 1 : 4.
Perpendiculars from an Internal Point
If perpendiculars of lengths P₁, P₂, and P₃ are drawn from any point inside an equilateral triangle to its three sides, the side 'a' is:
Side (a) = (2 / √3) * (P₁ + P₂ + P₃)
Isosceles Triangle (समद्विबाहु त्रिभुज)
An isosceles triangle has two equal sides ('a') and a base ('b'). The altitude from the vertex between the equal sides bisects the base.
Height (h): h = (1/2) * √(4a² - b²)
Area: Area = (b / 4) * √(4a² - b²)
Perimeter: Perimeter = 2a + b For problem-solving, it is often simpler to treat the isosceles triangle as two congruent right-angled triangles formed by the altitude, then use Area = (1/2) * base * height.
Scalene Triangle (विषमबाहु त्रिभुज)
A scalene triangle has three sides of different lengths (a, b, c).
General Area Formula: Area = (1/2) * Base * Height
Heron's Formula: When only the lengths of the three sides are known: Area = √[s(s - a)(s - b)(s - c)] where s is the semi-perimeter: s = (a + b + c) / 2
General Formulas for Inradius and Circumradius (For ANY Triangle)
These formulas are universal and apply to any triangle.
Inradius (r): r = Area / s (where 's' is the semi-perimeter)
Circumradius (R): R = (a * b * c) / (4 * Area) (where a, b, c are the side lengths)
Right-Angled Triangle (समकोण त्रिभुज)
A triangle with one angle of 90°. Let sides forming the right angle be 'a' and 'c' (perpendicular and base), and the hypotenuse be 'b'.
Area: Area = (1/2) * a * c
Perimeter: Perimeter = a + b + c
Pythagoras Theorem: a² + c² = b²
Length of Altitude to Hypotenuse (BD): BD = (a * c) / b
Special Cases for Inradius and Circumradius
Inradius (r): r = (a + c - b) / 2
Circumradius (R): R = b / 2
Right-Angled Isosceles Triangle
If the two non-hypotenuse sides are equal ('a'):
Hypotenuse: a√2
Area: (1/2)a²
Square and Rhombus
Square and Rhombus are important quadrilaterals frequently asked in SSC and other competitive exams. Both have all sides equal, but their angle and diagonal properties make them different in problem-solving.
Understanding their formulas, properties, and diagonal relationships helps solve area, perimeter, and geometry-based questions quickly.
1. The Square (वर्ग)
A square is a rhombus where every angle is 90°.
Properties: All four sides equal (a), all angles 90°.
Perimeter = 4a.
Diagonals (d) are equal, bisect each other at 90°, and are angle bisectors.
Length of a diagonal (d) = a√2.
Area: a² or (1/2)d².
Inradius (r): r = a / 2
Circumradius (R): R = a / √2
Key Ratios for a Square:
Ratio of Inradius to Circumradius (r : R): 1 : √2
Ratio of Areas of Incircle to Circumcircle: 1 : 2 This ratio is independent of the side length.
Shaded Area in a Square
For specific shaded patterns within a square of side 'a', such as the area between a square and its incircle, the shaded area can often be calculated directly:
Area of Shaded Region = (3/14)a² (Memory Tip: This formula is for specific shaded regions like the corners of a square after removing an inscribed circle.)
2. The Rhombus (समचतुर्भुज)
A rhombus is a quadrilateral with all four sides equal.
Properties of a Rhombus
Perimeter = 4a.
Area Formulas:
Using diagonals: Area = (1/2) * d₁ * d₂
Using base and height: Area = Base × Height
Relation between Side and Diagonals: d₁² + d₂² = 4a²
Square vs. Rhombus
Here are the key differences between Square and Rhombus:
Property
Square
Rhombus
Sides
All four sides are equal.
All four sides are equal.
Angles
All angles are 90°.
Opposite angles are equal.
Diagonals
Equal in length.
Unequal in length (d₁ ≠ d₂).
Bisect at 90°
Diagonals bisect each other at 90°.
Diagonals bisect each other at 90°.
Angle Bisector
Diagonals are angle bisectors.
Diagonals are angle bisectors.
Rectangle and Parallelogram
Rectangle and Parallelogram are important quadrilaterals in Mensuration and Geometry. Many SSC questions are directly based on their properties, area formulas, diagonals, and relationships between sides and angles. With clear concepts, these questions become quick and scoring.
Property
Rectangle (आयत)
Parallelogram (समांतर चतुर्भुज)
Sides
Opposite sides are equal (L, B).
Opposite sides are equal (L, B).
Perimeter
2(L + B)
2(L + B)
Angles
All angles are 90°.
Opposite angles are equal.
Diagonals
Equal in length.
Unequal in length.
Area
Length × Breadth (L × B)
Base × Corresponding Height
Parallelogram and Rectangle Properties
Rectangle Diagonals: Equal in length, Diagonal = √(L² + B²).
Using Triangles: Area = 2 × Area(scalene triangle formed by diagonal and two sides).
Area of Paths
Questions on Area of Paths are very common in SSC and other competitive exams. These problems involve finding the area of a pathway (road/border) built inside or outside a given geometric figure such as a rectangle, square, or circle.
Case
Description
Formula for Area of the Path
1. Path Around the Rectangle
Path of width d outside.
Area = 2d(L + B + 2d)
2. Path Inside the Rectangle
Path of width d inside.
Area = 2d(L + B - 2d)
3. Paths Along Length & Breadth
Two crossing paths of width d.
Area = d(L + B - d)
(Memory Tip: For a path around (outside) the rectangle, use a plus sign: + 2d. For a path inside the rectangle, use a minus sign: - 2d.)
Trapezium (Samlamb Chaturbhuj)
A trapezium is a quadrilateral with one pair of opposite sides being parallel.
Area Formula: Area = (1/2) × (Sum of parallel sides) × (Height) Area = (1/2) × (a + b) × h
Perimeter: Sum of all four sides.
Isosceles Trapezium
An isosceles trapezium has equal non-parallel sides. Its diagonals are equal.
Solution to Isosceles Trapezium Problem
Given Area 176, parallel sides 4x : 7x, height h = (2/11) * (11x) = 2x.
Area = (1/2)(4x+7x)(2x) = 11x². So, 11x² = 176 -> x² = 16 -> x = 4.
Parallel sides: 16 and 28. Height: 8.
For diagonal: Drop perpendiculars. Base segments are (28-16)/2 = 6.
Diagonal forms a right-angled triangle with height 8 and base (16+6) = 22.
A sector is a portion of a circle enclosed by two radii and an arc, forming an angle θ at the center.
Key Formulas for Sectors and Arcs:
Area of a Sector (A): A = (θ / 360) * πr²
Length of an Arc (L): L = (θ / 360) * 2πr
Area of a Sector using Arc Length: A = (1/2) * L * r (very important for direct calculation)
Quadrant of a Circle
A quadrant is a sector where the central angle θ = 90°.
Area of a Quadrant: (1/4)πr²
Perimeter of a Quadrant: (1/2)πr + 2r (arc length + two radii)
(Memory Tip: For the same perimeter, a shape with more sides (or continuous curve) encloses a larger area. Area(Circle) > Area(Square) > Area(Equilateral Triangle)).
Cylinder and Cone
A cylinder is a three-dimensional solid having two parallel circular bases connected by a curved surface.
A cone is a solid with a circular base and a single vertex (apex).
Feature
Cylinder
Cone
Bases
Two circular bases
One circular base + apex
Height
h
h
Radius
r
r
Slant Height (l)
Not applicable
l = √(h² + r²)
Volume (V)
V = πr²h
V = (1/3)πr²h
Curved Surface Area (CSA)
CSA = 2πrh
CSA = πrl
Total Surface Area (TSA)
TSA = 2πr(r + h)
TSA = πr(r + l)
(When to use CSA: For painting, wrapping, or coloring a surface, calculate CSA.)
Conceptual Relationship: Volume of Cylinder vs. Cone
If a cylinder and a cone have the same base radius (r) and same height (h), the volume of the cylinder is exactly three times the volume of the cone.
Volume_Cone = (1/3) * Volume_Cylinder.
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Mensuration FAQs
Q1: What are the primary classifications of triangles?
A1: Triangles are classified based on sides (equilateral, isosceles, scalene) and angles (acute, right-angled, obtuse).
Q2: How do you calculate the area of an equilateral triangle?
A2: The area of an equilateral triangle with side 'a' is given by the formula (√3 / 4) * a².
Q3: What is the relationship between the inradius and circumradius of an equilateral triangle?
A3: The ratio of the inradius (r) to the circumradius (R) in an equilateral triangle is 1:2.
Q4: How do you find the area of a rhombus given its diagonals?
A4: The area of a rhombus is calculated as (1/2) * d₁ * d₂, where d₁ and d₂ are the lengths of its diagonals.
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