Geometry
🔒 Log in to trackLines and angles, triangles and their four centres, similarity and BPT, Pythagoras, quadrilaterals, polygons and circles. The single largest advanced-maths block in CGL (3–5 questions per shift), and nearly every question runs off a memorised angle result, a triplet, or a ratio rule.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
4 exam-level questions worked step by step.
73 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (73 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Angles on parallel lines with x
very commonTwo angles on parallel lines are written with x, e.g. (5x − 40)° and (3x + 20)°.
Same-side (co-interior) angles add to 180°. Matching or alternate angles are equal.
Solve for x, then put it back to get the angle that was asked.
Two co-interior angles between parallel lines are (5x − 40)° and (3x + 20)°. The smaller angle is:
Co-interior angles add to 180°
(5x − 40) + (3x + 20) = 180, so 8x = 200 and x = 25
Angles: 5 × 25 − 40 = 85° and 3 × 25 + 20 = 95°
Smaller angle = 85°
Complement and supplement
very common"The supplement of an angle is 3 times its complement."
Complement = 90° − angle. Supplement = 180° − angle.
Write the sentence as one equation and solve.
The supplement of an angle is three times its complement. Find the angle.
Let the angle be x
180 − x = 3 × (90 − x)
180 − x = 270 − 3x, so 2x = 90
Angle = 45°
Angle at a triangle's special centre
very commonA triangle with a named centre (incentre I, circumcentre O or orthocentre H) and angle A given.
Incentre: angle BIC = 90° + half of A. Circumcentre: angle BOC = 2 × A.
Orthocentre: angle BHC = 180° − A.
In triangle ABC, ∠A = 50°. If O is the circumcentre, find ∠BOC.
O is the circumcentre
Angle at the centre = 2 × angle at the corner
2 × 50° = 100°
Centroid cuts the median 2 : 1
common"G is the centroid and median AD is 15 cm. Find AG."
The centroid cuts each median in the ratio 2 : 1, longer part at the corner.
So AG is ⅔ of the median and GD is ⅓.
The median AD of a triangle is 15 cm and G is the centroid. Find AG.
Ratio 2 : 1 makes 3 equal parts
1 part = 15 ÷ 3 = 5 cm
AG = 2 parts = 10 cm (GD = 5 cm)
Similar triangles: sides vs areas
very commonTwo similar triangles; perimeters or sides are given and an area is asked, or the reverse.
Sides and perimeters go by the ratio r. Areas go by r × r.
From areas back to sides, take the square root.
The areas of two similar triangles are 25 and 81 sq cm; the smaller perimeter is 40 cm. Find the larger perimeter.
Area ratio = 25 : 81
Side ratio = √25 : √81 = 5 : 9
Perimeter ratio is also 5 : 9
Larger perimeter = 40 × 9 ÷ 5 = 72 cm
Parallel line inside a triangle
common"DE is parallel to BC" inside triangle ABC, or D and E are midpoints, and a length is missing.
A parallel line cuts both sides in the same ratio: AD : DB = AE : EC.
If D and E are midpoints, DE is half of BC.
In triangle ABC, DE is parallel to BC with AD = 3, DB = 5, AE = 6. Find EC.
AD : DB = AE : EC
3 : 5 = 6 : EC
3 becomes 6 (doubled), so 5 becomes 10
EC = 10 cm
Right triangle hidden in a story
very commonA ladder on a wall, two poles with a rope between the tops, a walk north then east, or a diagonal.
Draw the right triangle. The ladder, rope or diagonal is the longest side.
Square the two short sides, add, then take the square root.
Two poles 6 m and 11 m high stand 12 m apart on level ground. Find the distance between their tops.
Height difference = 11 − 6 = 5 m
Short sides: 12 m and 5 m
12² + 5² = 144 + 25 = 169
Distance = √169 = 13 m
Regular polygon: angle and sides
commonAn interior or exterior angle of a regular polygon is given; asks the number of sides.
Exterior angle = 180° − interior angle.
Number of sides = 360° ÷ exterior angle.
Each interior angle of a regular polygon is 150°. Find the number of sides.
Exterior angle = 180° − 150° = 30°
Sides = 360° ÷ 30°
= 12
Rhombus diagonals or cyclic quadrilateral
very commonDiagonals of a rhombus with the side or perimeter asked, or a cyclic quadrilateral with one angle given.
Rhombus: the diagonals cross at right angles and cut each other in half, so use the half-diagonals as two short sides.
Cyclic quadrilateral: opposite angles add to 180°.
The diagonals of a rhombus are 10 cm and 24 cm. Find its perimeter.
Half-diagonals: 5 and 12
Side = √(25 + 144) = √169 = 13
Perimeter = 4 × 13 = 52 cm
Circle: chord and distance from centre
very commonA chord and its distance from the centre, tangents, or angles at the centre and on the edge.
Chord: half the chord, the distance and the radius make a right triangle.
Angle at the centre is twice the angle at the edge.
A chord 16 cm long lies in a circle of radius 10 cm. Find its distance from the centre.
Half the chord = 8 cm
Right triangle: 8² + d² = 10²
d² = 100 − 64 = 36
Distance = √36 = 6 cm