Geometry Notes for Competitive Exams

Geometry is a high-weightage area in SSC CGL, CHSL, CPO and CDS, and it also supports mensuration and coordinate questions in other exams. In UPSC CSAT, geometry appears through figure-based reasoning, areas and simple properties of triangles and circles. Unlike algebra, geometry needs you to "see" relations in a figure. The best approach is to learn a limited set of theorems very well, draw a rough figure for every question and mark the given data on it before calculating.

Quick Facts: Geometry at a Glance

• Angle sum of a triangle: 180°. Exterior angle = sum of the two opposite interior angles.
• Sum of interior angles of an n-sided polygon: (n − 2) × 180°.
• Pythagoras theorem: In a right triangle, hypotenuse² = base² + perpendicular².
• Four centres of a triangle: Centroid, Incentre, Circumcentre and Orthocentre.
• Circle: Angle in a semicircle is 90°; a tangent is perpendicular to the radius at the point of contact.

1. Lines and Angles

When a transversal cuts two parallel lines: corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles add up to 180°. These three facts solve most "find x" questions in figures with parallel lines.

Example: Two angles are supplementary and one is 40° more than the other. Then x + x + 40 = 180, so x = 70°. The angles are 70° and 110°.

2. Triangles: Basic Properties

• Sum of any two sides is greater than the third side; the difference of any two sides is less than the third side.
• The side opposite the larger angle is longer.
• Equilateral triangle (side a): height = (√3/2)a; area = (√3/4)a².
• Isosceles triangle: angles opposite equal sides are equal.
• Area = ½ × base × height; or by Heron's formula √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter.
• Area can also be found as ½ × a × b × sin C when two sides and the included angle are known.

Example: Can 3 cm, 4 cm and 8 cm form a triangle? 3 + 4 = 7, which is less than 8. So no.

Example: Find the area of a triangle with sides 13, 14 and 15. s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 sq units.

3. Four Centres of a Triangle

• In an equilateral triangle, all four centres coincide.
• In a right triangle, the circumcentre is the midpoint of the hypotenuse and the orthocentre is at the right-angle vertex. Circumradius = hypotenuse ÷ 2.
• The centroid divides the triangle into 6 small triangles of equal area.
• For an equilateral triangle of side a: inradius = a/(2√3), circumradius = a/√3, so circumradius = 2 × inradius.
• Inradius of any triangle = Area ÷ semi-perimeter; for a right triangle, r = (a + b − c)/2.

Example: In triangle ABC, ∠A = 70°. If I is the incentre, ∠BIC = 90° + 35° = 125°. If H is the orthocentre, ∠BHC = 180° − 70° = 110°.

4. Congruence and Similarity

Congruent triangles are equal in all respects. Tests: SSS, SAS, ASA, AAS and RHS.

Similar triangles have the same shape but may differ in size. Tests: AA, SSS (sides in proportion) and SAS (one angle equal and including sides in proportion).

• Ratio of areas of similar triangles = (ratio of corresponding sides)².
• Basic Proportionality Theorem (Thales): A line parallel to one side of a triangle divides the other two sides in the same ratio.
• Angle bisector theorem: The internal bisector of angle A divides BC in the ratio AB : AC.
• Midpoint theorem: The line joining midpoints of two sides is parallel to the third side and half of it.
• Apollonius theorem: AB² + AC² = 2(AD² + BD²), where AD is a median.

Example: Two similar triangles have sides in the ratio 3 : 5. Their areas are in the ratio 9 : 25.

5. Pythagoras Theorem and Triplets

In a right triangle, (hypotenuse)² = (base)² + (perpendicular)². Remember common Pythagorean triplets to save time:

• (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (9, 40, 41)
• Multiples also work: (6, 8, 10), (9, 12, 15), (10, 24, 26).

If c² < a² + b², the triangle is acute; if c² > a² + b², it is obtuse (c being the longest side).

Example: A 13 m ladder reaches a window 12 m high. Distance of the foot from the wall = √(169 − 144) = 5 m.

6. Circles

• The perpendicular from the centre to a chord bisects the chord.
• Equal chords are equidistant from the centre.
• Angle subtended by an arc at the centre is twice the angle at any point on the remaining circle.
• Angles in the same segment are equal.
• Angle in a semicircle is 90°.
• Opposite angles of a cyclic quadrilateral add up to 180°.
• A tangent is perpendicular to the radius at the point of contact.
• Two tangents from an external point are equal in length.
• Tangent-secant theorem: PT² = PA × PB.
• Intersecting chords: If chords AB and CD meet at P inside the circle, PA × PB = PC × PD.
• Length of direct common tangent = √[d² − (r1 − r2)²]; transverse common tangent = √[d² − (r1 + r2)²], where d is the distance between centres.

Example: A chord of length 16 cm is 6 cm from the centre. Radius = √(8² + 6²) = 10 cm.

Example: Two circles of radii 8 cm and 3 cm have centres 13 cm apart. Direct common tangent = √(169 − 25) = 12 cm.

7. Quadrilaterals

In a rhombus, side² = (d1/2)² + (d2/2)². Example: diagonals 16 and 12 give side = √(64 + 36) = 10.

Remember that the figure formed by joining the midpoints of the sides of any quadrilateral is a parallelogram, and its area is half that of the original quadrilateral.

8. Polygons

• Sum of interior angles = (n − 2) × 180°.
• Sum of exterior angles of any convex polygon = 360°.
• Each exterior angle of a regular polygon = 360°/n.
• Number of diagonals = n(n − 3)/2.
• Area of a regular hexagon of side a = (3√3/2)a², which is six equilateral triangles.

Example: Each interior angle of a regular polygon is 140°. Exterior angle = 40°, so n = 360/40 = 9 sides.

Common Mistakes to Avoid

1. Not drawing a figure and trying to solve mentally.
2. Mixing up incentre, circumcentre and orthocentre angle formulas.
3. Using side ratio instead of its square for area of similar triangles.
4. Assuming a figure is drawn to scale. Exam figures are often not.
5. Forgetting that a tangent makes 90° with the radius, which is the key step in most tangent questions.

Preparation Strategy

• Keep a theorem sheet with one solved example under each theorem.
• Solve 25 questions per sub-topic: lines, triangles, circles and polygons.
• Revise triplets and centre formulas every week, and analyse SSC and CDS previous year questions to see which theorems repeat.
• In the exam, if a geometry question takes more than 90 seconds, mark it and come back later.

Geometry becomes easy when you stop memorising answers and start recognising the theorem behind each figure. A neat rough figure, correctly labelled, solves half the question.

Geometry in Different Exams

SSC CGL asks theorem-based questions on triangle centres, circles and tangents. CDS tests similar ideas with more calculation. UPSC CSAT keeps geometry simple and mixes it with reasoning, such as counting triangles in a figure or comparing areas. Match your depth of practice to your target exam.

Related Notes

• Mensuration Notes: Area, Volume and Perimeter
• Ratio and Proportion Notes
• Number System Notes
• All Quantitative Aptitude Notes

Frequently Asked Questions

What is the sum of angles of a triangle?

The sum of the interior angles of any triangle is 180°.

What is the centroid of a triangle?

The centroid is the point where the three medians meet. It divides each median in the ratio 2 : 1 from the vertex.

What is the difference between incentre and circumcentre?

The incentre is where angle bisectors meet and is the centre of the inscribed circle. The circumcentre is where perpendicular bisectors meet and is equidistant from all vertices.

Where is the circumcentre of a right-angled triangle?

It lies at the midpoint of the hypotenuse, so the circumradius is half the hypotenuse.

What are Pythagorean triplets?

Sets of three whole numbers that satisfy a² + b² = c², such as (3, 4, 5), (5, 12, 13) and (8, 15, 17).

What is the angle in a semicircle?

The angle in a semicircle is always 90°.

What is a cyclic quadrilateral?

A quadrilateral whose vertices lie on a circle. Its opposite angles add up to 180°.

How do you find the number of diagonals in a polygon?

Use n(n − 3)/2, where n is the number of sides.

What is the ratio of areas of similar triangles?

It equals the square of the ratio of their corresponding sides.

What is the sum of exterior angles of a polygon?

The sum of exterior angles of any convex polygon is always 360°.

Is geometry asked in UPSC CSAT?

Yes, but at a basic level, usually through figures, areas and simple properties rather than advanced theorems.

What is the best way to prepare geometry?

Learn key theorems with one example each, draw a figure for every question and practise previous year questions regularly.