Comprehensive Geometry Cheat Sheet - IIM CAT 2026

Chapter 1: 2D Shapes, Area, and Perimeter

Understanding 2D plane geometry is essential for mastering the Quantitative Aptitude section of IIM CAT 2026. This chapter reviews core formulas for triangles, circles, quadrilaterals, and regular polygons.

1.1 Triangles & Heron's Formula

For any triangle with base \(b\) and height \(h\), or side lengths \(a\), \(b\), and \(c\):

\[\text{Area} = \frac{1}{2} b h = \sqrt{s(s-a)(s-b)(s-c)}\]

where the semi-perimeter is defined as \(s = \frac{a+b+c}{2}\) (Heron's Formula).

1.2 Right-Angled Triangles & Pythagorean Theorem

For a right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\):

\[a^2 + b^2 = c^2 \quad \text{(Pythagorean Theorem)}\]

1.3 Circles, Arcs, and Sectors

Given a circle of radius \(r\) and central angle \(\theta\) in radians:

• Area: \(\text{Area} = \pi r^2\)
• Circumference: \(\text{Circumference} = 2\pi r\)
• Arc Length (\(s\)): \(s = r\theta\)
• Sector Area: \(\text{Sector Area} = \frac{1}{2}r^2\theta\)

1.4 Quadrilaterals & Regular Polygons

Parallelogram:

\[\text{Area} = b h\]

Trapezoid (Trapezium):

\[\text{Area} = \frac{a + b}{2} \cdot h\]

where \(a\) and \(b\) are parallel side lengths, and \(h\) is the perpendicular height.

Regular Polygon (\(n\)-sided, side length \(a\)):

\[\text{Area} = \frac{n a^2}{4 \tan(\pi / n)}\]

\[\text{Interior Angle} = \frac{(n-2)180^\circ}{n}\]

Chapter 2: 3D Surface Area and Volume

Three-dimensional mensuration frequently tests candidates on volume optimization, surface area ratios, and composite solids in CAT Quantitative Aptitude.

2.1 Spheres and Cylinders

Sphere:

\[V = \frac{4}{3}\pi r^3, \quad A = 4\pi r^2\]

Cylinder:

\[V = \pi r^2 h, \quad A_{\text{total}} = 2\pi r(r + h)\]

2.2 Cones, Prisms, and Regular Pyramids

Right Circular Cone:

\[V = \frac{1}{3}\pi r^2 h, \quad A_{\text{lateral}} = \pi r l \quad \text{where } l = \sqrt{r^2 + h^2}\]

Rectangular Prism (Cuboid):

\[V = l w h, \quad A = 2(lw + lh + wh)\]

Regular Pyramid:

\[V = \frac{1}{3} B h \quad (B = \text{Base Area})\]

Chapter 3: Key Circle Theorems

Geometry questions involving circles often turn on angle-subtending properties, cyclic properties, or secant/tangent power relationships.

3.1 Inscribed Angle & Thales's Theorems

• Inscribed Angle Theorem: An angle \(\theta\) inscribed in a circle is half of the central angle \(2\theta\) subtending the same arc.
• Thales's Theorem: An angle inscribed in a semicircle is always a right angle (\(90^\circ\)).

3.2 Cyclic Quadrilaterals, Intersecting Chords & Tangent-Secant Theorem

• Cyclic Quadrilateral: Opposite angles sum to \(180^\circ\) (\(A + C = 180^\circ\)).
• Intersecting Chords Theorem: For two chords \(AB\) and \(CD\) intersecting at point \(P\), \(PA \cdot PB = PC \cdot PD\).
• Tangent-Secant Theorem: For a tangent segment \(PT\) and secant line passing through \(A\) and \(B\) from external point \(P\), \(PT^2 = PA \cdot PB\).

CAT Exam PYQ Analysis & Topic Weightage

Geometry and Mensuration constitute approximately 20% to 25% of the Quantitative Aptitude section in the CAT exam (typically 4-6 questions out of 22). Based on recent NTA and IIM exam patterns, Triangles and Circle Theorems carry the highest weightage.

• Triangles & Trigonometry: 2-3 Questions per slot (Focus on similarity, acute/obtuse conditions, and coordinate connections).
• Circles & Cyclic Polygons: 1-2 Questions per slot (Focus on power of a point, tangent-secant properties).
• Mensuration 3D: 1 Question per slot (Focus on cones, sphere-cylinder combinations, and ratio of volumes).

Frequently Asked Questions (People Also Ask)

What is the weightage of Geometry in IIM CAT 2026?

Geometry and Mensuration usually account for 4 to 6 questions out of 22 in the CAT Quantitative Aptitude section, making it one of the top three highest-yielding modules alongside Algebra and Arithmetic.

What is Heron's Formula and when should I use it?

Heron's Formula is \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \frac{a+b+c}{2}\). Use it when all three side lengths of a non-right triangle are known and calculating the perpendicular height is inconvenient.

What is the Tangent-Secant Theorem in Circle Geometry?

The Tangent-Secant Theorem states that if a tangent segment \(PT\) and a secant line segment \(PAB\) are drawn from an external point \(P\) to a circle, then \(PT^2 = PA \cdot PB\).