Quick Theory Recap

Geometry questions in Railway exams are built around five core ideas: angles, triangles, quadrilaterals, circles and mensuration.

• Angles on a straight line add to 180°, around a point to 360°; when two lines are parallel, a transversal creates equal corresponding angles and interior allied angles that are supplementary.
• Triangle facts: sum of interior angles = 180°; exterior angle = sum of opposite interior angles; Pythagoras (a² + b² = c²) is tested almost every year; congruence rules (SSS/SAS/ASA/RHS) and similarity (AAA) are used to set-up proportional sides.
• Special triangles—45°-45°-90° sides 1:1:√2 and 30°-60°-90° sides 1:√3:2—save calculation time.
• Quadrilaterals: parallelogram opposite sides/angles equal, diagonals bisect; rhombus diagonals perpendicular; rectangle diagonals equal; square combines all.
• Circle: angle at centre = 2 × angle at circumference subtended by same arc; angle in semicircle = 90°; tangents from an external point equal; radius ⟂ tangent.
• Mensuration: areas (rectangle l×b, triangle ½bh, circle πr², trapezium ½(a+b)h) and volumes (cylinder πr²h, cone ⅓πr²h, sphere 4/3πr³) are frequently asked in 1-mark problems. Memorise the formulas and look for symmetry or “cut-and-paste” tricks to avoid lengthy calculations.

Practice Set – 25 MCQs

1. Two angles of a triangle are 42° and 68°. The third angle is
   A. 50°
   B. 60°
   C. 70°
   D. 80°

2. If each side of a square is increased by 10%, its area increases by
   A. 10%
   B. 21%
   C. 25%
   D. 30%

3. The diagonal of a 9 cm × 12 cm rectangle is
   A. 10 cm
   B. 12 cm
   C. 13 cm
   D. 15 cm

4. In a rhombus, one diagonal is 8 cm and the other is 6 cm. Its area is
   A. 12 cm²
   B. 24 cm²
   C. 36 cm²
   D. 48 cm²

5. The radius of a circle is 7 cm. Approximate circumference is (π = 22/7)
   A. 22 cm
   B. 44 cm
   C. 66 cm
   D. 88 cm

6. A right triangle with legs 15 cm and 20 cm has hypotenuse
   A. 23 cm
   B. 24 cm
   C. 25 cm
   D. 26 cm

7. An angle is 30° more than its complement. The angle is
   A. 30°
   B. 45°
   C. 60°
   D. 75°

8. The sum of exterior angles of a 12-sided polygon is
   A. 180°
   B. 360°
   C. 720°
   D. 1800°

9. A ladder 5 m long reaches 4 m high. How far is the foot from the wall?
   A. 2 m
   B. 3 m
   C. 4 m
   D. 5 m

10. Area of a semicircle of diameter 14 cm is (π = 22/7)
    A. 77 cm²
    B. 154 cm²
    C. 308 cm²
    D. 616 cm²

11. Two concentric circles have radii 5 cm and 12 cm. The length of a chord of the larger circle tangent to the smaller one is
    A. 10 cm
    B. 12 cm
    C. 16 cm
    D. 22 cm

12. In ΔABC, DE ∥ BC with AD = 3 cm, DB = 6 cm, AE = 2 cm. Then EC is
    A. 3 cm
    B. 4 cm
    C. 5 cm
    D. 6 cm

13. The area of an equilateral triangle of side 6 cm is
    A. 9√3 cm²
    B. 18√3 cm²
    C. 24 cm²
    D. 36 cm²

14. A cylindrical tank of radius 1 m and height 2 m can hold (π = 3.14)
    A. 6.28 L
    B. 628 L
    C. 6280 L
    D. 62800 L

15. The number of diagonals in a hexagon is
    A. 6
    B. 9
    C. 12
    D. 15

16. A circle is inscribed in a square of side 14 cm. The area inside the square but outside the circle is (π = 22/7)
    A. 42 cm²
    B. 84 cm²
    C. 126 cm²
    D. 154 cm²

17. The angles of a quadrilateral are in ratio 2:3:4:5. The largest angle is
    A. 120°
    B. 135°
    C. 150°
    D. 160°

18. A train wheel of diameter 70 cm makes 200 revolutions. Distance moved is
    A. 220 m
    B. 440 m
    C. 880 m
    D. 1760 m

19. In a right triangle, one angle is 34°. The other acute angle is
    A. 56°
    B. 66°
    C. 90°
    D. 146°

20. A parallelogram has sides 10 cm and 6 cm and one diagonal 8 cm. Its area is
    A. 24 cm²
    B. 30 cm²
    C. 48 cm²
    D. 60 cm²

21. A cone has height 24 cm and slant height 25 cm. Its volume is (π = 22/7)
    A. 1232 cm³
    B. 1848 cm³
    C. 2464 cm³
    D. 3080 cm³

22. A sphere of radius 3 cm is melted and recast into a cylinder of radius 1 cm. The height of the cylinder is
    A. 12 cm
    B. 24 cm
    C. 36 cm
    D. 48 cm

23. The angle subtended by a chord at the centre is 110°. The angle subtended at the circumference is
    A. 55°
    B. 110°
    C. 220°
    D. 125°

24. A trapezium has parallel sides 8 cm and 14 cm and height 5 cm. Its area is
    A. 55 cm²
    B. 60 cm²
    C. 65 cm²
    D. 70 cm²

25. A clock’s minute hand is 7 cm long. The area swept in 30 minutes is (π = 22/7)
    A. 77 cm²
    B. 154 cm²
    C. 231 cm²
    D. 308 cm²

Quick Shortcuts & Tips

1. 3-4-5 family: Any multiple (6-8-10, 9-12-15…) is a right triangle—no need to square-root.
2. π approximations: 22/7 gives 1% error; for quick multiply, π ≈ 3.1 and add 1% extra if choices are close.
3. Angle hunts: Whenever parallel lines appear, mark all corresponding/alternate angles equal in one go—saves re-reading.
4. Area by symmetry: Rhombus/kite area = ½ d₁ d₂; remember diagonals are perpendicular so you can split into four right triangles.
5. Revolutions to distance: Distance = number of revolutions × π × diameter—don’t forget to convert cm → m at the end.
6. Cone & sphere volume ratio: Cone is ⅓ of cylinder; sphere is 4/3 πr³—keep 4, 3, 2 (for hemisphere) in mind to guess factor quickly.
7. Clock hands: Minute hand sweeps 360° in 60 min → 6° per min; hour hand 0.5° per min; relative speed 5.5°/min—useful for angle-between-hands questions.
8. Eliminate absurd units: Area can never be in cm, volume never in cm²—spot unit mismatch in options to reject 1-2 choices instantly.

Keep a formula palm-card: area & perimeter on one side, volume on the other; 30-second glance before exam sets visual memory.