In this guide
Geometry questions in RRB NTPC usually test one fact at a time. If you know the fact, the question takes seconds; if you do not, no amount of calculation will rescue you. So this chapter is about knowing a short list of facts properly, including why they are true, because the "why" is what helps you recall them under pressure.
Everything here is Class 9–10 level. Lengths and areas are in the next chapter, mensuration.
Lines and angles
- Angles on a straight line add up to 180°; angles around a point add up to 360°.
- Complementary angles add to 90°; supplementary angles add to 180°.
- Vertically opposite angles are equal.
- With two parallel lines and a transversal: alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°.
Triangles
- The three angles add up to 180°.
- An exterior angle equals the sum of the two opposite interior angles. Why: the exterior angle and the adjacent interior angle make 180°, and so do all three interior angles. Remove the shared angle from both and what remains is equal.
- The sum of any two sides is greater than the third side.
- The largest angle is opposite the longest side.
Pythagoras and triplets
In a right triangle, hypotenuse² = base² + height². Triplets let you skip the square roots:
| Triplet | Multiples worth knowing |
|---|---|
| 3, 4, 5 | 6, 8, 10; 9, 12, 15; 12, 16, 20 |
| 5, 12, 13 | 10, 24, 26 |
| 8, 15, 17 | 16, 30, 34 |
| 7, 24, 25 | 14, 48, 50 |
Special points and similarity
| Idea | Fact to remember |
|---|---|
| Centroid | Where the medians meet; divides each median in the ratio 2 : 1 from the vertex |
| Right triangle | The circumcentre is the midpoint of the hypotenuse, so circumradius = hypotenuse ÷ 2 |
| Equilateral triangle | Height = (√3/2) × side |
| Midpoint theorem | The line joining midpoints of two sides is parallel to the third side and half of it |
| Similar triangles | Ratio of areas = square of the ratio of corresponding sides |
Quadrilaterals and polygons
- The angles of any quadrilateral add up to 360°.
- Sum of interior angles of an n-sided polygon = (n − 2) × 180°. Why: from one corner you can draw diagonals that split the polygon into (n − 2) triangles, each with 180°.
- Sum of exterior angles of any convex polygon = 360°, so each exterior angle of a regular n-gon = 360° ÷ n.
- Interior angle + exterior angle = 180° at every corner.
| Polygon | Sides | Sum of interior angles | Each angle (regular) | Each exterior angle |
|---|---|---|---|---|
| Pentagon | 5 | 540° | 108° | 72° |
| Hexagon | 6 | 720° | 120° | 60° |
| Octagon | 8 | 1,080° | 135° | 45° |
| Decagon | 10 | 1,440° | 144° | 36° |
Circles
- The angle in a semicircle is 90°.
- The angle at the centre is twice the angle at the circumference on the same arc.
- Angles in the same segment are equal.
- In a cyclic quadrilateral, opposite angles add up to 180°.
- The perpendicular from the centre to a chord bisects the chord. So (half-chord)² + distance² = radius².
- A tangent is perpendicular to the radius at the point of contact, and tangents from an external point are equal in length. So tangent² = (distance from centre)² − radius².
Why the tangent formula works: the radius, the tangent and the line to the centre form a right triangle with the right angle at the point of contact. Pythagoras does the rest.
Worked questions at NTPC level
Q1. The angles of a triangle are in the ratio 2 : 3 : 4. Find them.
2x + 3x + 4x = 180, so x = 20. Angles: 40°, 60° and 80°.
Q2. An exterior angle of a triangle is 110°, and one of the opposite interior angles is 45°. Find the other opposite interior angle.
110 − 45 = 65°.
Q3. Each exterior angle of a regular polygon is 24°. How many sides does it have?
360 ÷ 24 = 15 sides.
Q4. A point is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from the point.
Tangent² = 169 − 25 = 144. Tangent = 12 cm (a 5-12-13 triplet).
Q5. A chord of length 16 cm is drawn in a circle of radius 10 cm. How far is it from the centre?
Half-chord = 8. Distance² = 100 − 64 = 36. Distance = 6 cm.
Q6. In a cyclic quadrilateral ABCD, ∠A = 70° and ∠B = 95°. Find ∠C and ∠D.
Opposite angles add to 180°: ∠C = 110° and ∠D = 85°.
Common mistakes
- Using (n − 2) × 180 for exterior angles. The exterior angles always total 360°.
- Forgetting the right angle at a tangent. Almost every tangent question is a hidden Pythagoras question.
- Using the full chord in Pythagoras. Halve it first.
- Mixing up complement (90°) and supplement (180°).
- Taking the side ratio as the area ratio for similar triangles. Square it.
Practice set
- Two angles of a triangle are 45° and 75°. Find the third.
- A right triangle has legs of 5 cm and 12 cm. Find the hypotenuse.
- Find each interior angle of a regular octagon.
- Each interior angle of a regular polygon is 140°. How many sides does it have?
- A point is 10 cm from the centre of a circle of radius 6 cm. Find the tangent length.
- A chord of length 24 cm is drawn in a circle of radius 13 cm. How far is it from the centre?
- A right triangle has legs of 9 cm and 12 cm. Find its circumradius.
- A median of a triangle is 12 cm long. How far is the centroid from the vertex on that median?
Answers:
- 60°. 180 − 120.
- 13 cm. 5-12-13 triplet.
- 135°. 1,080 ÷ 8.
- 9 sides. Exterior angle = 40°; 360 ÷ 40.
- 8 cm. 100 − 36 = 64.
- 5 cm. Half-chord 12; 169 − 144 = 25.
- 7.5 cm. Hypotenuse 15 (9-12-15); half of it.
- 8 cm. The centroid divides the median 2 : 1, so 2/3 of 12.
What to do next
- Write out the triplet table and the polygon table from memory until both come out correct twice in a row.
- Solve 25 mixed geometry questions and note beside each which single fact it used.
- Draw a rough figure for every circle question, marking right angles at tangents and at chord midpoints.
- Continue with mensuration and trigonometry, which build on these triangles.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Railway Recruitment Boards website .
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