Geometry Cheat Sheet - IIM CAT 2026

PYQ Analysis & Weightage for CAT 2026

Geometry and Mensuration constitute approximately 20% to 25% of the Quantitative Aptitude section in the IIM CAT exam. Historically, CAT tests conceptual application over plain formula memorization. Expect roughly 4 to 6 questions directly from triangles, circles, coordinate geometry, and 3D solids in CAT 2026. Mastering core properties, area formulas, and geometric theorems provides high-yield returns for aspirants aiming for a 99+ percentile.

Chapter 1: Basic Angles & Lines

Understanding angle relationships forms the bedrock of planar geometry and transversal problems in competitive exams.

• Complementary Angles: Two angles are complementary when their sum is \(\alpha + \beta = 90^\circ\).
• Supplementary Angles: Two angles are supplementary when their sum is \(\alpha + \beta = 180^\circ\).
• Vertically Opposite Angles: When two lines intersect, the vertically opposite angles are strictly equal.
• Parallel Lines Transversal: When a transversal intersects parallel lines, corresponding angles, alternate interior angles, and alternate exterior angles are equal.

Chapter 2: Triangles & Trigonometric Area Formats

Triangles are the most heavily tested topic under geometry in CAT. Candidates must be comfortable with right-angled properties, Heron's formula, and equilateral triangle shortcuts.

Core Formulas

• Angle Sum Property: \(\alpha + \beta + \gamma = 180^\circ\).
• General Area: \[A = \frac{1}{2}bh\]
• Heron's Formula: \[A = \sqrt{s(s-a)(s-b)(s-c)}\] where semi-perimeter \(s = \frac{a+b+c}{2}\).
• Equilateral Triangle Area: \[A = \frac{\sqrt{3}}{4}a^2\]
• Pythagorean Theorem: In a right-angled triangle with hypotenuse \(c\), \[a^2 + b^2 = c^2\]

Chapter 3: Polygons & Interior Angle Properties

Polygon properties frequently feature in data sufficiency and quantitative reasoning problems. Focus on diagonal calculations and regular polygon interior/exterior angle relationships.

• Sum of Interior Angles: \((n-2) \times 180^\circ\) for an \(n\)-sided polygon.
• Each Interior Angle (Regular Polygon): \[\frac{(n-2)180^\circ}{n}\]
• Sum of Exterior Angles: Always equals \(360^\circ\) regardless of the number of sides.
• Number of Diagonals: \[\frac{n(n-3)}{2}\]

Chapter 4: Circles & Inscribed Angle Theorems

Circle geometry problems in CAT revolve around chord properties, arcs, central versus inscribed angles, and sector areas.

• Circumference: \(C = 2\pi r\)
• Area: \(A = \pi r^2\)
• Arc Length: \(s = r\theta\) (where angle \(\theta\) is measured in radians).
• Sector Area: \[A = \frac{1}{2}r^2\theta\]
• Inscribed Angle Theorem: An angle inscribed in a circle is half of the central angle that subtends the same arc.

Chapter 5: 3D Solids Mensuration

Mensuration questions test three-dimensional spatial reasoning involving cylinders, cones, and spheres.

• Cylinder Volume: \(V = \pi r^2 h\)
• Cylinder Total Surface Area: \(S = 2\pi r h + 2\pi r^2\)
• Cone Volume: \[V = \frac{1}{3}\pi r^2 h\]
• Sphere Volume: \[V = \frac{4}{3}\pi r^3\]
• Sphere Surface Area: \(S = 4\pi r^2\)

Chapter 6: Coordinate Geometry Fundamentals

Coordinate geometry bridges algebraic formulas with geometric visualization. Key concepts include distance, midpoints, and linear equations.

• Distance Formula: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]
• Midpoint Formula: \[M = \bigl(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\bigr)\]
• Slope (m): \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
• Slope-Intercept Form: \(y = mx + b\)

Frequently Asked Questions (People Also Ask)

Q1: How many questions appear from Geometry in CAT Quantitative Aptitude?

Typically, 4 to 6 questions out of 22 questions in the Quant section belong to Geometry and Mensuration, making it one of the highest-weightage topics alongside Arithmetic and Algebra.

Q2: What is the formula for the number of diagonals in an n-sided polygon?

The formula for finding the number of diagonals in a polygon with \(n\) sides is \[\frac{n(n-3)}{2}\]

Q3: How is sector area calculated when angle is given in radians?

When the angle \(\theta\) is in radians, the area of a sector of a circle with radius \(r\) is given by \[A = \frac{1}{2}r^2\theta\]