Geometry

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Geometry — Key Facts for CLAT Quantitative Techniques • Triangle: Sum of interior angles = $180°$. Area = $\dfrac{1}{2} \times \text{base} \times \text{height}$. • Pythagorean Theorem (Right-angled triangle): $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.

• Pythagorean triplets: $(3,4,5), (5,12,13), (7,24,25), (8,15,17), (9,40,41)$ • Circle: Area = $\pi r^2$; Circumference = $2\pi r$. • Rectangle: Area = $l \times w$; Perimeter = $2(l+w)$; Diagonal = $\sqrt{l^2 + w^2}$. • Trapezium: Area = $\dfrac{1}{2}(a+b) \times h$ where $a,b$ are parallel sides. • Similar Triangles: Corresponding angles equal → corresponding sides in the same ratio. • Angle Bisector Theorem: The bisector of an angle divides the opposite side in the ratio of the adjacent sides.

⚡ Exam Tip: In CLAT Quantitative Techniques, most geometry questions involve word problems where a diagram is NOT provided. Draw a quick figure on your rough sheet — it immediately clarifies relationships between shapes and angles.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Geometry — CLAT Study Guide

Core Concept: CLAT geometry tests applied measurement and spatial reasoning. Questions involve finding areas, perimeters, angles, and side lengths of 2D shapes embedded in word problems. No theorems are required beyond those listed above, but their application must be precise.

Key Formulas — Areas:

Shape	Area Formula
Triangle	$\frac{1}{2}bh$
Square	$a^2$ (side²)
Rectangle	$l \times w$
Parallelogram	$b \times h$
Rhombus	$\frac{1}{2} \times d_1 \times d_2$ (diagonals product)
Trapezium	$\frac{1}{2}(a+b) \times h$
Circle	$\pi r^2$ (use $\pi = 22/7$ or 3.14 as directed)

Pythagorean Triplets — Memorise These:

Triplet	Verification
3, 4, 5	$9+16=25$
5, 12, 13	$25+144=169$
7, 24, 25	$49+576=625$
8, 15, 17	$64+225=289$
9, 40, 41	$81+1600=1681$
11, 60, 61	$121+3600=3721$

Worked Example 1 (Right Triangle Application): A ladder 15 metres long leans against a wall. The foot of the ladder is 9 m from the wall. How high up the wall does the ladder reach?

• By Pythagoras: $h^2 + 9^2 = 15^2$ → $h^2 + 81 = 225$ → $h^2 = 144$ → $h = 12$ m

Worked Example 2 (Area Word Problem): A rectangular park is 60 m long and 40 m wide. A path 2 m wide runs around the inside of the park. Find the area of the path.

• Total park area = $60 \times 40 = 2400$ m²
• Inner rectangle dimensions: $(60-4) \times (40-4) = 56 \times 36 = 2016$ m² (subtract 2 m from each side twice)
• Path area = $2400 - 2016 = 384$ m²

Worked Example 3 (Circle): The radius of a circular garden is 7 m. A path 1 m wide surrounds it. Find the area of the path. (Use $\pi = 22/7$)

• Garden area = $\pi \times 7^2 = 154$ m²
• Outer radius = $7 + 1 = 8$ m; Outer area = $\pi \times 8^2 = 201.14$ m²
• Path area = $201.14 - 154 = 47.14$ m² ≈ $47.14$ m² or exactly: $\pi(64-49) = 15\pi = 47.14$ m²

Common Student Mistakes: Forgetting that the path reduces the inner rectangle by 4 m (2 m each side) not 2 m, mixing up the formulas for rhombus vs parallelogram, and using diameter instead of radius in circle area calculations.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Geometry — Comprehensive CLAT Notes

Theoretical Foundation: CLAT geometry is practical measurement geometry. The emphasis is on applying area, perimeter, and volume formulas correctly within word problem scenarios. Law contexts include property boundary calculations, inheritance division, lease area computations, and court hall seating arrangements.

Triangle Properties:

1. Sum of Angles: $\angle A + \angle B + \angle C = 180°$
2. Exterior Angle Theorem: Exterior angle = sum of opposite interior angles
3. Triangle Inequality: Sum of any two sides > third side; Difference of any two sides < third side
4. Similar Triangles: If two triangles are similar (all angles equal), then: $$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$$

Worked Example — Similar Triangles: A pole 6 m tall casts a shadow 8 m long. At the same time, a building casts a shadow 24 m long. How tall is the building?

• Sun’s angle is the same → triangles are similar
• $\frac{6}{8} = \frac{h}{24}$ → $h = \frac{6 \times 24}{8} = 18$ m

Polygon Interior Angles:

• Sum of interior angles of an $n$-sided polygon = $(n-2) \times 180°$
• Each interior angle of a regular $n$-gon = $\dfrac{(n-2) \times 180°}{n}$
• Example: Regular hexagon (n=6): $(6-2) \times 180/6 = 120°$ per interior angle

Arc and Sector:

• Arc length = $\dfrac{\theta}{360°} \times 2\pi r$ (where $\theta$ is in degrees)
• Area of sector = $\dfrac{\theta}{360°} \times \pi r^2$

3D Geometry — Mensuration:

Shape	Volume	Surface Area
Cube	$a^3$	$6a^2$
Cuboid	$l \times w \times h$	$2(lw+lh+wh)$
Cylinder	$\pi r^2 h$	$2\pi r(h+r)$
Cone	$\frac{1}{3}\pi r^2 h$	$\pi r(l+r)$ where $l = \sqrt{r^2 + h^2}$
Sphere	$\frac{4}{3}\pi r^3$	$4\pi r^2$

Worked Example — Cylinder: A cylindrical tank has radius 3.5 m and height 10 m. Find the volume in litres. (1 m³ = 1000 litres)

• Volume = $\pi \times 3.5^2 \times 10 = \frac{22}{7} \times 12.25 \times 10 = 385$ m³
• In litres = $385,000$ litres

CLAT PYQ Pattern (2019-2024):

• 2024: 8-10 Quantitative Techniques questions total; geometry = 2-3 questions (circle area, triangle similar, polygon angles)
• 2023: Pythagorean theorem question + area of combined shapes
• 2022: Cylinder volume + triangle angle sum
• Most common: Circle-based problems (area, circumference, sector) — appear in 60% of geometry questions
• Difficulty: Moderate — numbers are manageable and calculators are not needed
• Traps: Units mismatch (cm² vs m²), forgetting that diameter = 2r, mixing up “around” (perimeter) vs “inside” (area)

Preparation Strategy: Memorise all area and volume formulas. Practice unit conversion (1 hectare = 10,000 m²; 1 litre = 1000 cm³). For Pythagorean triplets, remember that scaling a triplet (e.g., 6, 8, 10 from 3, 4, 5) also works. Focus on circle and triangle geometry — these appear most frequently.

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