Quick Overview
Geometry in CAT tests visual clarity, formula recall, and ability to combine shapes. Focus on: circle theorems (angles & chords), triangle properties (similarity, Pythagoras, sine/cosine rules), polygons (interior/exterior angles, diagonals), and mensuration (areas, perimeters, volumes). Diagram-first approach: always sketch, label, and derive step-by-step.
Circles
Key concepts (with diagram):
Important: Angle in a semicircle = 90°. Tangent ⟂ radius at point of contact. Equal chords subtend equal angles at center. Power of a point & cyclic quadrilateral properties are high-yield.
Worked examples
Example: If chord AB subtends 60° at centre, arc AB length = (60/360)×2πr = (πr)/3.
Triangles
Key: Pythagoras for right triangles. Similarity → ratio of sides equal. Area formula
Δ = 1/2 × base × height, Heron's formula Δ = √(s(s−a)(s−b)(s−c)).Important theorems
- Sine rule:
a/sinA = b/sinB = c/sinC. - Cosine rule:
a^2 = b^2 + c^2 − 2bc cosA. - Area via sides: Heron's formula.
Polygons
Interior angle of n-gon =
(n−2)×180°/n. Number of diagonals = n(n−3)/2. Area formulas depend on regularity: regular polygon area = (1/4) n s^2 cot(π/n).Mensuration (Areas, Perimeters, Volumes)
Volume examples: cylinder
V = πr^2 h; sphere V = 4/3 π r^3; cone V = 1/3 π r^2 h. Convert units carefully.Worked example
Example: Area of shaded semicircle of radius r = (1/2)πr^2. If inside a square of side r, remaining area = r^2 − (1/2)πr^2.
50 Practice MCQs — Geometry (Answers highlighted)
High-yield geometry MCQs using the diagrams and formulae above. Each question aims to test core theorem knowledge, quick diagram reasoning, and mensuration conversions.
Q1. Angle in a semicircle is:
Q2. Two radii of a circle subtend an arc of 120°. Central angle = ?
Q3. If chord subtends equal angles at centre, chords are:
Q4. Tangent from external point touches circle at T. Radius OT at T is:
Q5. In triangle with sides 3,4,5 area = ?
Q6. Sum of interior angles of hexagon = ?
Q7. Diagonals of rectangle are:
Q8. Area of equilateral triangle side a = ?
Q9. Interior angle of regular octagon = ?
Q10. If radius r, area of circle = ?
Q11. Right triangle legs 5 and 12 hypotenuse = ?
Q12. If two similar triangles have ratio of sides 2:3, ratio of areas = ?
Q13. Area of trapezium with parallel sides a and b and height h = ?
Q14. Number of diagonals in a decagon = ?
Q15. Circumference of circle with diameter d = ?
Q16. In right triangle, median to hypotenuse equals:
Q17. Area of sector with angle θ (radians) = ?
Q18. For an isosceles triangle with sides (a,a,b), altitude to base b = ? (in terms of a and b)
Q19. If regular pentagon side s, interior angle = ?
Q20. Area of rectangle length l width w = ?
Q21. If two chords are equal, they subtend equal angles at:
Q22. Perimeter of regular hexagon with side s = ?
Q23. The orthocentre of an acute triangle lies:
Q24. Area of right triangle with legs a and b = ?
Q25. A circle with diameter d has area = ?
Q26. If triangle sides 7,24,25, is it right-angled?
Q27. Volume of cylinder radius r height h = ?
Q28. If interior angles of triangle are in arithmetic progression, what are they (in degrees)?
Q29. If square has diagonal d, side = ?
Q30. If two angles of triangle are 45° and 45°, triangle is:
Q31. Area of sector with central angle 180° (half circle) = ?
Q32. If polygon has n sides, sum of exterior angles = ?
Q33. In a circle, the perpendicular from centre to chord bisects:
Q34. If regular triangle (equilateral) has perimeter P, area = ? (in terms of P)
Q35. If two chords are equidistant from centre, they are:
Q36. Volume of sphere radius r = ?
Q37. If triangle sides are proportional 3:4:5 it is:
Q38. Area of regular n-gon with side s approximates to?
Q39. If a triangle has sides 8,15,17, area = ?
Q40. If arc length = rπ/2, central angle is:
Q41. If triangle area = 30 and base = 5, height = ?
Q42. If polygon interior angle = 120°, number of sides = ?
Q43. If circle radius 7, circumference ≈ ? (use π=22/7)
Q44. If triangle is equilateral side 6, area = ?
Q45. If chord length equals diameter, chord is:
Q46. Area of ring between two concentric circles r2>r1 = ?
Q47. If polygon has 8 sides, each exterior angle = ?
Q48. If a cone radius 3 and height 4, volume = ? (π omitted)
Q49. If two circles touch externally, distance between centres = ?
Q50. Best approach when geometry question looks complex:
Practice & Test-day Tips
- Always draw a clear labeled diagram. Mark right angles, equal sides, parallel lines, and given lengths.
- Memorize key circle theorems and triangle area formulas; practice common mensuration conversions (cm↔m).
- Use symmetry and similarity to reduce multi-shape problems to basic units.
- For numeric answers, estimate to eliminate wrong options quickly.