SSC Geometry Questions with Solutions (Triangles, Circles and Polygons)

Geometry carries four to six questions in SSC CGL Tier 1 and Tier 2, so it is one of the most important chapters in Quantitative Aptitude. The chapter is theorem-based, which means that if you learn about 25 important results, most questions become direct. This practice paper has 30 geometry questions with answers and short solutions covering triangles, polygons, circles, tangents and triangle centres.

Practice Paper at a Glance

What Is Geometry? Basic Concept in Simple Words

Geometry deals with the properties of lines, angles, triangles, circles and other shapes. In SSC exams, geometry questions mainly test the following areas.

• Lines and angles: parallel lines, transversal, co-interior and alternate angles.
• Triangles: angle sum, exterior angle, similarity, congruence and the Pythagoras theorem.
• Triangle centres: centroid, incentre, circumcentre and orthocentre.
• Polygons: interior and exterior angles and the number of diagonals.
• Circles: chords, tangents, cyclic quadrilaterals and angles in segments.

Important Theorems, Formulas and Shortcut Tricks for SSC Geometry

• Angle sum of a triangle = 180 degrees. Exterior angle = sum of the two opposite interior angles.
• Polygon: sum of interior angles = (n - 2) x 180. Each interior angle of a regular polygon = (n - 2) x 180 / n. Each exterior angle = 360 / n. Diagonals = n(n - 3)/2.
• Pythagoras: hypotenuse^2 = base^2 + perpendicular^2. Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
• Similar triangles: ratio of areas = square of the ratio of corresponding sides.
• Basic proportionality theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.
• Angle bisector theorem: the bisector divides the opposite side in the ratio of the other two sides.
• Centroid divides each median in the ratio 2 : 1 from the vertex.
• Incentre: angle BOC = 90 + A/2. Circumcentre: angle BOC = 2A. Orthocentre: angle BOC = 180 - A.
• Circle: angle at the centre is twice the angle at the circumference. Angle in a semicircle = 90. Opposite angles of a cyclic quadrilateral add up to 180.
• Tangents: tangent is perpendicular to the radius at the point of contact. Tangents from an external point are equal. Direct common tangent = square root of (d^2 - (R - r)^2). Transverse common tangent = square root of (d^2 - (R + r)^2).
• Inradius of a right triangle = (base + perpendicular - hypotenuse) / 2.

Set 1: Triangles and Polygons (Q1–Q10)

Q1. Two angles of a triangle are 50 degrees and 60 degrees. Find the third angle.
Answer: 70 degrees
Solution: Third angle = 180 - (50 + 60) = 70 degrees.

Q2. The exterior angle of a triangle is 110 degrees and one of the opposite interior angles is 45 degrees. Find the other opposite interior angle.
Answer: 65 degrees
Trick: Exterior angle = sum of the two opposite interior angles, so the other angle = 110 - 45 = 65 degrees.

Q3. Find the sum of the interior angles of a hexagon.
Answer: 720 degrees
Solution: (n - 2) x 180 = 4 x 180 = 720 degrees.

Q4. Find each interior angle of a regular octagon.
Answer: 135 degrees
Solution: (8 - 2) x 180 / 8 = 1,080 / 8 = 135 degrees.

Q5. Each exterior angle of a regular polygon is 36 degrees. Find the number of its sides.
Answer: 10
Trick: n = 360 / 36 = 10 sides.

Q6. Find the number of diagonals of a decagon.
Answer: 35
Trick: Diagonals = n(n - 3)/2 = 10 x 7 / 2 = 35.

Q7. The two sides of a right triangle are 6 cm and 8 cm. Find the hypotenuse.
Answer: 10 cm
Solution: Hypotenuse = square root of (36 + 64) = 10 cm.

Q8. The hypotenuse of a right triangle is 25 cm and one side is 7 cm. Find the other side.
Answer: 24 cm
Solution: Other side = square root of (625 - 49) = square root of 576 = 24 cm.

Q9. The angles of a triangle are in the ratio 2 : 3 : 4. Find the largest angle.
Answer: 80 degrees
Solution: Sum of parts = 9. One part = 20 degrees. Largest = 4 x 20 = 80 degrees.

Q10. The vertex angle of an isosceles triangle is 40 degrees. Find each base angle.
Answer: 70 degrees
Solution: Base angles = (180 - 40) / 2 = 70 degrees.

Set 2: Circles, Chords and Tangents (Q11–Q20)

Q11. An arc subtends an angle of 80 degrees at the centre of a circle. Find the angle it subtends at any point on the remaining circumference.
Answer: 40 degrees
Trick: Angle at the circumference is half the angle at the centre = 40 degrees.

Q12. What is the angle in a semicircle?
Answer: 90 degrees
Solution: The angle in a semicircle is always a right angle.

Q13. One angle of a cyclic quadrilateral is 85 degrees. Find the opposite angle.
Answer: 95 degrees
Trick: Opposite angles add up to 180, so 180 - 85 = 95 degrees.

Q14. A point P is 13 cm from the centre O of a circle of radius 5 cm. Find the length of the tangent from P.
Answer: 12 cm
Solution: Tangent length = square root of (13^2 - 5^2) = square root of 144 = 12 cm.

Q15. A chord of length 16 cm is in a circle of radius 10 cm. Find the distance of the chord from the centre.
Answer: 6 cm
Solution: Half chord = 8. Distance = square root of (100 - 64) = 6 cm.

Q16. PA and PB are tangents from an external point P to a circle with centre O. If angle APB = 50 degrees, find angle AOB.
Answer: 130 degrees
Trick: Angle AOB + angle APB = 180 (quadrilateral with two right angles). So angle AOB = 130 degrees.

Q17. The angle between a tangent and a chord at the point of contact is 35 degrees. Find the angle in the alternate segment.
Answer: 35 degrees
Solution: By the alternate segment theorem, the angle is equal to 35 degrees.

Q18. Two circles of radii 9 cm and 4 cm touch externally and the distance between their centres is 13 cm. Find the length of the direct common tangent.
Answer: 12 cm
Trick: Length = square root of (13^2 - (9 - 4)^2) = square root of (169 - 25) = 12 cm.

Q19. Two circles with radii 5 cm and 3 cm have centres 17 cm apart. Find the length of the transverse common tangent.
Answer: 15 cm
Trick: Length = square root of (17^2 - (5 + 3)^2) = square root of (289 - 64) = 15 cm.

Q20. Find the radius of the incircle of a right triangle with sides 6 cm, 8 cm and 10 cm.
Answer: 2 cm
Trick: Inradius = (6 + 8 - 10) / 2 = 2 cm.

Set 3: Similar Triangles, Triangle Centres and Mixed Questions (Q21–Q30)

Q21. The length of a median of a triangle is 18 cm. How far is the centroid from the vertex along this median?
Answer: 12 cm
Solution: The centroid divides the median in the ratio 2 : 1 from the vertex, so the distance = 2/3 x 18 = 12 cm.

Q22. Triangles ABC and DEF are similar and their corresponding sides are in the ratio 2 : 3. If the area of triangle ABC is 20 sq cm, find the area of triangle DEF.
Answer: 45 sq cm
Trick: Ratio of areas = 4 : 9. Area of DEF = 20 x 9/4 = 45 sq cm.

Q23. In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD = 3 cm, DB = 5 cm and AC = 16 cm, find AE.
Answer: 6 cm
Solution: AD / DB = AE / EC = 3 / 5. So AE = 16 x 3/8 = 6 cm.

Q24. In triangle ABC, AD is the bisector of angle A. If AB = 6 cm, AC = 9 cm and BC = 10 cm, find BD.
Answer: 4 cm
Trick: BD / DC = AB / AC = 6 / 9 = 2 / 3. BD = 10 x 2/5 = 4 cm.

Q25. Find the area of an equilateral triangle of side 8 cm.
Answer: 16 root 3 sq cm (about 27.7 sq cm)
Solution: Area = (root 3 / 4) x side^2 = (root 3 / 4) x 64 = 16 root 3 sq cm.

Q26. O is the incentre of triangle ABC and angle A = 60 degrees. Find angle BOC.
Answer: 120 degrees
Trick: Angle BOC = 90 + A/2 = 90 + 30 = 120 degrees.

Q27. O is the circumcentre of triangle ABC and angle A = 40 degrees. Find angle BOC.
Answer: 80 degrees
Trick: Angle BOC = 2 x angle A = 80 degrees.

Q28. Each interior angle of a regular polygon is 150 degrees. Find the number of sides.
Answer: 12
Solution: Exterior angle = 30 degrees. Number of sides = 360 / 30 = 12.

Q29. In a right-angled triangle ABC with the right angle at A, AD is perpendicular to BC. If BD = 4 cm and DC = 9 cm, find AD.
Answer: 6 cm
Trick: AD^2 = BD x DC = 36, so AD = 6 cm.

Q30. Two parallel lines are cut by a transversal. The co-interior angles are 3x and 2x. Find the smaller angle.
Answer: 72 degrees
Solution: Co-interior angles add up to 180: 5x = 180, x = 36. Smaller angle = 2 x 36 = 72 degrees.

Common Mistakes in Geometry

• Mixing the formulas for the incentre, circumcentre and orthocentre angles.
• Using the direct common tangent formula for the transverse tangent (the sign inside the bracket differs).
• Forgetting that the angle at the centre is double the angle at the circumference.
• Applying Pythagoras to a triangle that is not right-angled.
• Taking the ratio of areas as equal to the ratio of sides in similar triangles (it is the square).
• Not drawing a clear figure before solving.

How to Practice These 30 Questions

1. Make a one-page sheet of theorems and revise it daily.
2. Draw a neat diagram for every question and mark the given values.
3. Solve Set 1 in 8 minutes, Set 2 in 12 minutes and Set 3 in 12 minutes.
4. Keep a notebook for tricky questions and revisit it every week.
5. Attempt previous year SSC geometry questions after finishing this paper.

Related Topics to Study Next

• Mensuration Questions
• Trigonometry Questions
• SSC Algebra Questions
• SSC Previous Year Papers

Frequently Asked Questions

How many geometry questions come in SSC CGL?

Generally four to six questions come from geometry in the Tier 1 paper, and Tier 2 has more. Triangles and circles are the most important topics.

What are the most important geometry theorems for SSC?

The angle sum and exterior angle of a triangle, Pythagoras theorem, basic proportionality theorem, angle bisector theorem, the circle theorems (angle at centre, semicircle, cyclic quadrilateral and tangents) and the angle formulas for triangle centres.

What is the formula for the interior angle of a regular polygon?

Each interior angle of a regular polygon with n sides is (n - 2) x 180 / n degrees. Each exterior angle is 360 / n degrees.

What is the angle at the incentre, circumcentre and orthocentre?

At the incentre, angle BOC = 90 + A/2. At the circumcentre, angle BOC = 2A. At the orthocentre, angle BOC = 180 - A.

What is the alternate segment theorem?

The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment of the circle.

What is the formula for the length of common tangents?

Direct common tangent = square root of (d^2 - (R - r)^2) and transverse common tangent = square root of (d^2 - (R + r)^2), where d is the distance between the centres and R and r are the radii.

How do I solve similar triangle questions quickly?

Identify the corresponding sides, write the ratio of sides and remember that the ratio of areas is the square of the ratio of sides. The ratio of perimeters is the same as the ratio of sides.

What is the centroid and how does it divide a median?

The centroid is the point where the three medians meet. It divides every median in the ratio 2 : 1, with the longer part towards the vertex.

How do I remember geometry theorems?

Group them by topic, draw one small figure for each theorem, revise them daily and apply them in practice questions so that they stay in memory.

How should I prepare geometry for SSC exams?

Learn the important theorems, solve topic-wise sets every day, practise drawing figures quickly and take previous year papers to understand the exam pattern.