Geometry (RRB)

Triangles:

• Sum of interior angles = 180°.
• Exterior angle = sum of opposite interior angles.
• Triangle inequality: each side < sum of the other two.
• Area = ½ × base × height. With sides a,b,c and semiperimeter s = (a+b+c)/2: Area = √(s(s−a)(s−b)(s−c)) (Heron's formula).
• Equilateral triangle (side a): area = (√3/4)a², height = (√3/2)a.
• Right triangle: a² + b² = c² (Pythagoras). Common Pythagorean triples to memorise: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29.
• 30-60-90 triangle: sides in ratio 1 : √3 : 2. 45-45-90: 1 : 1 : √2.

Quadrilaterals & polygons:

• Square (side a): area = a², diagonal = a√2.
• Rectangle (l, b): area = lb, diagonal = √(l²+b²).
• Parallelogram: area = base × height.
• Rhombus: area = ½ × d₁ × d₂ (diagonals).
• Trapezium: area = ½ × (sum of parallel sides) × height.
• Regular polygon of n sides: interior angle = (n−2) × 180°/n. Sum = (n−2)×180°.

Circle (radius r, diameter d = 2r):

• Circumference = 2πr; Area = πr².
• Arc length = (θ/360°) × 2πr (θ in degrees).
• Sector area = (θ/360°) × πr² = ½r²θ (θ in radians).
• Chord of length 2r·sin(θ/2) subtends angle θ at centre.
• Angle in semicircle = 90° (Thales' theorem).
• Tangent-radius = 90° at the point of contact.

3-D mensuration:

• Cube (side a): V = a³, TSA = 6a², diagonal = a√3.
• Cuboid (l,b,h): V = lbh, TSA = 2(lb+bh+hl), diagonal = √(l²+b²+h²).
• Cylinder (r, h): V = πr²h, TSA = 2πr(r+h), CSA = 2πrh.
• Cone (r, h, l=slant): V = ⅓πr²h, CSA = πrl, l = √(r²+h²).
• Sphere (r): V = (4/3)πr³, SA = 4πr².
• Hemisphere: V = (2/3)πr³, TSA = 3πr².

Example 1 — Pythagorean triple recognition:
A ladder of length 25 m leans against a wall. The foot is 7 m from the wall. How high does the ladder reach?
Method: 7-?-25 is a triple. Recall 7-24-25. Answer: 24 m.

Example 2 — Cylinder volume:
A cylindrical tank of radius 7 m and height 10 m. Volume = π×7²×10 = 22/7 × 49 × 10 = 1540 m³.

Example 3 — Cone surface area:
Cone of base radius 6 cm and height 8 cm. Slant l = √(36+64) = √100 = 10 cm. CSA = π×6×10 = 60π cm² ≈ 188.4 cm².

Example 4 — Sector of a circle:
A pizza slice has radius 14 cm and central angle 90°. Area = (90/360) × π × 14² = ¼ × 22/7 × 196 = 154 cm².

Example 5 — Triangle altitude relation:
In a right triangle with legs 6 and 8, the hypotenuse is 10. The altitude to the hypotenuse = (product of legs) / hypotenuse = 48/10 = 4.8.

Example 6 — Sphere melted into wires:
A sphere of radius 6 cm is melted and drawn into a wire of radius 0.2 cm. Find the length.
Method: Volume conserved. (4/3)π × 6³ = π × (0.2)² × L ⟹ L = (4/3 × 216) / 0.04 = 288 / 0.04 = 7200 cm = 72 m.

Common traps:

• Confusing CSA (curved/lateral) with TSA (total). For a cylinder, TSA = CSA + 2 circle ends.
• Forgetting to square the radius when computing cone slant: l² = r² + h².
• π values: in RRB problems use 22/7 if the numbers divide cleanly, else use 3.14.