SSC CGL 2027 Geometry Questions Based on Triangles and Circles

Geometry is an important topic in the Quantitative Aptitude section of SSC CGL 2027. Questions based on triangles and circles test your understanding of angles, sides, theorems, area, circumference and geometric relationships.

Learning important formulas and practising standard question patterns can help candidates solve geometry problems with greater speed and accuracy. This article covers triangle properties, circle theorems, solved examples, shortcuts and practice MCQs for SSC CGL Tier 1 and Tier 2 preparation.

1. Important Triangle Properties for SSC CGL 2027

A triangle is a closed geometric figure with three sides, three vertices and three interior angles. Understanding its basic properties is essential for solving geometry questions.

• Angle sum property: The sum of the three interior angles of a triangle is 180°.
• Exterior angle property: An exterior angle equals the sum of the two opposite interior angles.
• Triangle inequality: The sum of any two sides of a triangle must be greater than the third side.
• Equal sides: Equal sides of a triangle have equal opposite angles.
• Equal angles: Equal angles have equal opposite sides.
• Right-angled triangle: The square of the hypotenuse equals the sum of the squares of the other two sides.

2. Types of Triangles

Triangles can be classified according to their sides or angles.

• Equilateral triangle: All three sides are equal, and each interior angle is 60°.
• Isosceles triangle: Two sides are equal, and the angles opposite those sides are equal.
• Scalene triangle: All three sides have different lengths.
• Acute-angled triangle: All three angles are less than 90°.
• Right-angled triangle: One angle is exactly 90°.
• Obtuse-angled triangle: One angle is greater than 90°.

3. Important Triangle Formulas

Memorise these formulas for quick calculations in SSC CGL geometry questions.

• Area of a triangle: ½ × Base × Height
• Perimeter: Sum of all three sides
• Right triangle: Hypotenuse² = Base² + Perpendicular²
• Equilateral triangle perimeter: 3 × Side
• Area of an equilateral triangle: (√3/4) × Side²
• Heron's formula: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2

Example 1: Two angles of a triangle are 55° and 65°. Find the third angle.

Solution:

Sum of angles = 180°

Third angle = 180° − (55° + 65°)

= 180° − 120°

Answer: 60°

Example 2: A right-angled triangle has sides of 6 cm and 8 cm forming the right angle. Find the hypotenuse.

Solution:

Hypotenuse² = 6² + 8²

= 36 + 64 = 100

Hypotenuse = √100

Answer: 10 cm

4. Similarity and Congruence of Triangles

Triangle similarity and congruence are useful concepts in geometry problems involving corresponding sides and angles.

Similar triangles: Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional.

Congruent triangles: Two triangles are congruent when they have the same shape and size. Their corresponding sides and angles are equal.

Common triangle congruence criteria include:

• SSS: Three corresponding sides are equal.
• SAS: Two corresponding sides and the included angle are equal.
• ASA: Two corresponding angles and the included side are equal.
• RHS: In right-angled triangles, the hypotenuse and one corresponding side are equal.

Example 3: Two similar triangles have corresponding sides in the ratio 2:3. If the smaller triangle has an area of 24 cm², find the area of the larger triangle.

Solution:

Ratio of areas = Square of the ratio of corresponding sides

Area ratio = 2²:3² = 4:9

Area of larger triangle = 24 × 9/4

Answer: 54 cm²

5. Important Circle Properties

A circle is a closed curve in which every point on the circumference is at the same distance from the centre.

• Radius: Distance from the centre to any point on the circle.
• Diameter: A chord passing through the centre; Diameter = 2 × Radius.
• Chord: A line segment joining two points on the circumference.
• Circumference: The total distance around the circle.
• Tangent: A line that touches a circle at exactly one point.
• Secant: A line that intersects a circle at two points.

6. Important Circle Formulas

• Circumference: 2πr or πd
• Area: πr²
• Diameter: 2r
• Length of a semicircular arc: πr
• Area of a semicircle: ½πr²
• Area of a quadrant: ¼πr²

Here, r represents the radius and d represents the diameter. Unless specified otherwise, π is commonly taken as 22/7 or 3.14, depending on the question.

Example 4: Find the circumference of a circle with a radius of 7 cm. Use π = 22/7.

Solution:

Circumference = 2πr

= 2 × 22/7 × 7

Answer: 44 cm

Example 5: Find the area of a circle with a radius of 14 cm. Use π = 22/7.

Solution:

Area = πr²

= 22/7 × 14 × 14

Answer: 616 cm²

7. Important Circle Theorems

Circle theorems are frequently useful when solving geometry questions involving angles, chords and tangents.

• Angle at the centre: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference.
• Angle in a semicircle: The angle subtended by a diameter at the circumference is 90°.
• Equal chords: Equal chords of the same circle subtend equal angles at the centre.
• Perpendicular from the centre: A perpendicular drawn from the centre to a chord bisects the chord.
• Tangent-radius theorem: The radius drawn to the point of contact is perpendicular to the tangent.
• Tangents from an external point: The lengths of two tangents drawn from the same external point to a circle are equal.

Example 6: An angle subtended by an arc at the circumference is 35°. Find the angle subtended by the same arc at the centre.

Solution:

Angle at the centre = 2 × Angle at the circumference

= 2 × 35°

Answer: 70°

Example 7: A tangent touches a circle at point P. If O is the centre, what is the angle between OP and the tangent at P?

Solution:

The radius drawn to the point of contact is perpendicular to the tangent.

Answer: 90°

8. Common Triangle and Circle Question Patterns

• Finding missing angles using the angle sum property of triangles.
• Calculating the hypotenuse or a missing side of a right-angled triangle.
• Finding areas of similar triangles using side ratios.
• Calculating the circumference or area of a circle.
• Finding angles using the centre and circumference theorem.
• Solving questions involving tangents, chords and radii.
• Applying multiple theorems to find an unknown angle or length.

9. Practice Questions for SSC CGL 2027

Attempt these multiple-choice questions to test your understanding of triangles and circles.

Question 1: Two angles of a triangle are 48° and 72°. Find the third angle.

• A) 50°
• B) 60°
• C) 70°
• D) 80°

Question 2: A right-angled triangle has perpendicular sides of 9 cm and 12 cm. Find the hypotenuse.

• A) 13 cm
• B) 14 cm
• C) 15 cm
• D) 16 cm

Question 3: Find the area of a circle with a radius of 7 cm. Use π = 22/7.

• A) 44 cm²
• B) 88 cm²
• C) 154 cm²
• D) 308 cm²

Question 4: An angle at the circumference subtended by an arc is 40°. Find the angle subtended by the same arc at the centre.

• A) 20°
• B) 40°
• C) 80°
• D) 100°

Question 5: The radius of a circle is 14 cm. Find its circumference using π = 22/7.

• A) 44 cm
• B) 66 cm
• C) 88 cm
• D) 154 cm

Question 6: The corresponding sides of two similar triangles are in the ratio 3:5. What is the ratio of their areas?

• A) 3:5
• B) 6:10
• C) 9:25
• D) 27:125

Question 7: What is the angle between a radius and a tangent at the point of contact?

• A) 45°
• B) 60°
• C) 90°
• D) 180°

10. Answers with Explanations

1. B) 60°
Third angle = 180° − (48° + 72°) = 60°.

2. C) 15 cm
Hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15 cm.

3. C) 154 cm²
Area = πr² = 22/7 × 7 × 7 = 154 cm².

4. C) 80°
Angle at the centre = 2 × 40° = 80°.

5. C) 88 cm
Circumference = 2πr = 2 × 22/7 × 14 = 88 cm.

6. C) 9:25
The ratio of areas of similar triangles is the square of the ratio of corresponding sides: 3²:5² = 9:25.

7. C) 90°
The radius is perpendicular to the tangent at the point of contact.

11. Preparation Tips for SSC CGL 2027 Geometry

• Memorise important triangle and circle formulas.
• Practise angle-based questions and right-triangle calculations regularly.
• Understand circle theorems instead of relying only on memorisation.
• Draw a simple diagram whenever a question describes a geometric relationship.
• Revise common Pythagorean triples such as 3-4-5, 5-12-13 and 8-15-17.
• Solve timed practice sets and review errors after each mock test.

Conclusion

Triangles and circles are essential areas of geometry preparation for SSC CGL 2027. A clear understanding of angle properties, triangle formulas, circle measurements and important theorems can make many questions easier to solve.

Revise the formulas regularly, practise the solved examples and attempt MCQs under timed conditions to improve your accuracy and confidence in Quantitative Aptitu