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Mensuration 2D: Triangles and Circles Concepts & PYQs Notes

Mensuration 2D: Triangles and Circles Concepts & PYQs

This lecture covers the foundational concepts, classifications, and formulas of 2D mensuration focusing on triangles and circles. It explains triangle existence conditions, classification by sides and angles, Pythagorean triplets, sector area, and concentric circles. Several previous years' questions from exams like LDC, CET, REET, and Forest Guard are solved in detail.

Other State PSC
SSC
Teaching
Maths
English · 8 topics
Checked against the video on 21 Oct 2026

Made by AI from the YouTube video Mensuration 2D One Shot 💯 | Complete Chapter in One Class 🔥 | LDC/CET/REET 2026 (opens YouTube in a new tab) by MATHS BY JAIDEEP SIR, then checked against it. All credit for the teaching goes to the creator. Tap a topic’s time to confirm a fact in the video.

Mensuration 2D: Triangles and Circles Concepts & PYQs

English8 topics · 7 MCQs · 12 flashcardsWatch the video (opens YouTube in a new tab)

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In short

This lecture covers the foundational concepts, classifications, and formulas of 2D mensuration focusing on triangles and circles. It explains triangle existence conditions, classification by sides and angles, Pythagorean triplets, sector area, and concentric circles. Several previous years' questions from exams like LDC, CET, REET, and Forest Guard are solved in detail.

Introduction to Mensuration 2D

Mensuration 2D deals with two-dimensional geometric figures, calculating their perimeter, boundary, and area.
Key exam questions covered in this session are selected from previous years' exams including Rajasthan LDC, CET, REET, Police, and Patwar.
Systematic practice of formulas along with previous year questions (PYQs) ensures long-term retention of geometrical concepts.

Fundamental Properties of Triangles

A triangle is a closed figure bounded by three straight sides having three vertices and three angles.
The sum of all three interior angles of any triangle is always 180 degrees (Angle A + Angle B + Angle C = 180°).
Must knowIn any valid triangle, the sum of any two sides must always be strictly greater than the third side (a + b > c, b + c > a, c + a > b).
Must knowThe difference between any two sides of a triangle must always be strictly less than the third side (|a - b| < c).
A triangle cannot be formed with side lengths 1, 2, and 3 because 1 + 2 = 3, which fails the condition that the sum must exceed the third side.

Triangle Existence and Counting Possible Triangles

When two side lengths are given (a and b), the third side length (x) must satisfy the inequality: |a - b| < x < (a + b).
If two sides are 5 and 7, the third integer side must be greater than 2 and less than 12 (i.e., integers from 3 to 11, giving 9 possible triangles).
If two sides are 4 and 7, the third integer side must be greater than 3 and less than 11 (integers 4 to 10, giving 7 possible triangles).
Must knowTo find the total number of possible triangles with integer sides given two sides, use the direct shortcut: 2 × (smallest side) - 1.
- For sides 5 and 7: 2 × 5 - 1 = 9 possible triangles.
- For sides 4 and 7: 2 × 4 - 1 = 7 possible triangles.

Classification of Triangles by Sides and Angles

Triangles are classified based on sides into Equilateral (all 3 sides equal), Isosceles (2 sides equal), and Scalene (all 3 sides unequal).
Triangles are classified based on angles into Acute-angled (all angles < 90°), Right-angled (one angle = 90°), and Obtuse-angled (one angle > 90°).
In any triangle, the side opposite to the largest angle is always the longest side.
Must knowIf c is the longest side of a triangle, it is:
- Acute-angled if a² + b² > c²
- Right-angled if a² + b² = c²
- Obtuse-angled if a² + b² < c²

Condition	Triangle Type (by Angle)	Example Sides
a² + b² > c²	Acute-angled	4, 5, 6 (16 + 25 = 41 > 36)
a² + b² = c²	Right-angled	3, 4, 5 and 5, 12, 13
a² + b² < c²	Obtuse-angled	2, 3, 4 (4 + 9 = 13 < 16)

Equilateral, Right-Angled, Scalene, and Isosceles Triangles Formulas

An equilateral triangle has all three angles equal to 60°; its perimeter is 3 × side, height is (√3 / 2) × side, and area is (√3 / 4) × side².
A right-angled triangle obeys the Pythagoras theorem (Hypotenuse² = Base² + Perpendicular²), and its area is 1/2 × Base × Height.
Standard Pythagorean triplets to memorize include: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), and (20, 21, 29).
A scalene triangle's area is calculated using Heron's formula: √[s(s - a)(s - b)(s - c)], where semi-perimeter s = (a + b + c) / 2.
In an isosceles triangle with equal sides 'a' and base 'b', the altitude bisects the base at right angles; its area is (b / 4) × √(4a² - b²).

Triangle PYQs Solutions

LDC 2018: A right-angled triangle has height 12 cm and base 4 cm; Area = 1/2 × 4 × 12 = 24 cm².
Patwar 2016: A right-angled triangle with sides forming the right angle as a and b has area equal to 1/2 ab.
REET L-2 (27 Feb 2023): An equilateral triangle of side 3 cm has area = (√3 / 4) × 3² = (9√3 / 4) cm².
CET (10+2) 2023: If two triangles are similar and the ratio of corresponding sides is 3:5, the ratio of their areas is (3/5)² = 9:25.
CET (Graduate Level) 2023: If the ratio of areas of two equilateral triangles is 36:25, the ratio of their perimeters equals the ratio of their sides, which is √(36/25) = 6:5.
CET (10+2) 2023: Similar triangles ABC and PQR have areas 25 cm² and 36 cm²; with AB = 2.5 cm, PQ is found from AB/PQ = √(25/36) = 5/6, giving PQ = 3.0 cm.
REET L-2 2022: A right triangle has perpendicular sides of 15 cm and 8 cm; the hypotenuse is √(15² + 8²) = 17 cm, so Perimeter = 8 + 15 + 17 = 40 cm.
Clerk Gr. II 2016: The sides of a right triangle are three consecutive even integers; using the scaled triplet 2 × (3, 4, 5) = (6, 8, 10), Area = 1/2 × 6 × 8 = 24 cm².
Forest Guard 2022: An isosceles triangle with base 24 cm and area 192 cm² gives height = (192 × 2) / 24 = 16 cm; equal sides = √(12² + 16²) = 20 cm; Perimeter = 20 + 20 + 24 = 64 cm.
Forest Guard 2022: In an equilateral triangle with area 'a' and height 'b', a / b² = [(√3 / 4) × s²] / [(3 / 4) × s²] = 1 / √3.
CET (10+2) 2024: An isosceles triangle has perimeter 32 cm and equal sides equal to 5/6 of the base; sides are 10, 10, and 12 cm; height = √(10² - 6²) = 8 cm; Area = 1/2 × 12 × 8 = 48 cm².
CET (10+2) 2024: Triangle ABC has AB = BC and AC = √2 × AB, with perimeter 14 + 7√2 cm; solving gives equal side = 7 cm, and Area = 49/2 cm² (or 24.5 cm²).
LDC 2018: The sides of a right triangle in arithmetic progression are uniquely in the ratio 3, 4, 5.
CET (10+2) 2023: If height of a triangle is doubled (ratio 1 to 2) and base is reduced by 20% (ratio 5 to 4), the ratio of old area to new area is (5 × 1) : (4 × 2) = 5:8.

Circles: Properties, Sectors, and Annulus

A circle is the locus of points equidistant from a fixed center; radius (r) is half of the diameter (d = 2r).
Circumference of a circle C = 2πr, representing the distance covered in one complete rotation.
Area of a circle A = πr², where π is approximately 22/7 or 3.14.
Length of an arc subtending angle θ at the center: L = (2πrθ) / 360° = (πrθ) / 180°.
Area of a sector of angle θ: Area = (πr^2θ) / 360°.
Concentric circles have the same center but different radii; the area of the ring/annulus between them is π(R1² - R2²).
Semi-circle area is 1/2 πr²; its perimeter including the diameter is πr + 2r.

Radius (r)	Diameter (d)	Circumference (2πr)	Area (πr²)
7	14	44	154
14	28	88	616
21	42	132	1386
28	56	176	2464
35	70	220	3850

Circles PYQs Solutions

REET L-2 2023: Area of a circular plate is 154 cm²; using standard values, radius = 7 cm and circumference = 44 cm.
CET (10+2) 2023: Circumference is 56 cm; 2 × (22/7) × r = 56 gives r = 98/11 cm; Area = (22/7) × (98/11) × (98/11) = 2744 / 11 ≈ 249.45 cm².
REET L-2 2018: Diameter is 56 cm (radius = 28 cm); full circumference is 176 cm; the circumference of the half-circle is 176 / 2 = 88 cm.
LDC 2018: The ratio of circumferences of two circles is 2:3; the ratio of their areas is the square of the ratio of circumferences, which is 2² : 3² = 4:9.

Key terms

Circumference: The boundary distance of a circle, equal to 2πr
Sector: The region of a circle bounded by two radii and the arc between them
Concentric Circles: Circles sharing the exact same center point with different radii
Semi-perimeter: Half of the perimeter of a polygon, denoted as s = (a + b + c) / 2 for a triangle
Pythagorean Triplet: A set of three positive integers a, b, c satisfying a² + b² = c²

Formulas

Number of integer triangles with two sides given: 2 × (smallest side) - 1
Area of Equilateral Triangle: (√3 / 4) × side²
Height of Equilateral Triangle: (√3 / 2) × side
Area of Right-Angled Triangle: 1/2 × Base × Height
Heron's Formula: √[s(s - a)(s - b)(s - c)], where s = (a + b + c) / 2
Area of Isosceles Triangle: (b / 4) × √(4a² - b²)
Area of a Circle: πr²
Circumference of a Circle: 2πr
Length of Arc: (πrθ) / 180°
Area of Sector: (πr^2θ) / 360°
Area of Annulus / Ring: π(R1² - R2²)
Perimeter of Semicircle: πr + 2r

Exam tips and traps

Memorize the standard base circle values for radius 7: diameter = 14, circumference = 44, area = 154. If radius is scaled by k, circumference scales by k, and area scales by k².

Quick revision

Read these the night before the exam.

The sum of any two sides of a triangle must strictly exceed the third side.
The difference of any two sides of a triangle must be strictly less than the third side.
An equilateral triangle of side s has height (√3 / 2)s and area (√3 / 4)s².
If the ratio of sides of two similar triangles is a:b, the ratio of their areas is a²:b².
In right triangles, standard integer triplets are (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).
For circle radius 7 cm, the circumference is 44 cm and the area is 154 cm².
The ratio of the area of an equilateral triangle to the square of its height is always 1 / √3.
A right triangle whose sides are in arithmetic progression always has sides in the ratio 3:4:5.

These notes are made by AI from the video. Use the timestamps to check any important fact against the video before you rely on it.