Geometry Cheat Sheet - NDA

Welcome to the comprehensive, LibreTexts-style study guide for NDA Geometry. This reference manual consolidates essential geometric principles, 2D and 3D mensuration formulas, coordinate geometry equations, and transformation rules specifically tailored for the UPSC National Defence Academy (NDA) Mathematics examination.

Table of Contents

• NDA Geometry: Weightage & PYQ Analysis
• 1. Basic Angles & Lines
• 2. Triangles
• 3. Polygons
• 4. Circles
• 5. 3D Solids
• 6. Coordinate Geometry
• 7. Transformations
• People Also Ask (FAQ)

NDA Geometry: Weightage & PYQ Analysis

In the UPSC NDA Mathematics paper, Geometry and Mensuration (along with 2D/3D Coordinate Geometry) constitute roughly 15% to 20% of the total 120 questions. Mastering fundamental theorems, properties of polygons, coordinate transformations, and spatial mensuration allows candidates to secure high-accuracy marks efficiently during the exam.

1. Basic Angles & Lines

Understanding linear relationships and angle properties formed by intersecting and parallel lines forms the cornerstone of geometric reasoning.

• Supplementary Angles: Two angles \(\alpha\) and \(\beta\) are supplementary if their sum is \(180^{\circ}\): \[\alpha + \beta = 180^{\circ}\]
• Complementary Angles: Two angles \(\alpha\) and \(\beta\) are complementary if their sum is \(90^{\circ}\): \[\alpha + \beta = 90^{\circ}\]
• Vertically Opposite Angles: When two lines intersect, opposite angles are equal in measure.
• Parallel Lines & Transversals: When a transversal intersects parallel lines:
  • Alternate interior angles are equal.
  • Corresponding angles are equal.
  • Consecutive interior angles are supplementary (sum to \(180^{\circ}\)).

2. Triangles

Triangles represent the primary polygon evaluated in plane geometry problems under the NDA syllabus.

• Sum of Angles: For any triangle with interior angles \(\alpha\), \(\beta\), and \(\gamma\): \[\alpha + \beta + \gamma = 180^{\circ}\]
• General Area Formula: Given base \(b\) and height \(h\): \[A = \frac{1}{2} b h\]
• Heron's Formula: For sides \(a\), \(b\), and \(c\): \[A = \sqrt{s(s-a)(s-b)(s-c)}\] where semi-perimeter \(s = \frac{a+b+c}{2}\).
• Equilateral Triangle Area: For side length \(a\): \[A = \frac{\sqrt{3}}{4} a^2\]
• Pythagorean Theorem: In a right-angled triangle with hypotenuse \(c\) and legs \(a\), \(b\): \[a^2 + b^2 = c^2\]

3. Polygons

Closed 2D figures bounded by straight lines are governed by specific angle and diagonal formulas based on the number of sides \(n\).

• Sum of Interior Angles: \[S = (n-2) \times 180^{\circ}\]
• Each Interior Angle (Regular Polygon): \[\theta = \frac{(n-2)180^{\circ}}{n}\]
• Sum of Exterior Angles: For any convex polygon, the sum of exterior angles is always \(360^{\circ}\).
• Number of Diagonals: \[D = \frac{n(n-3)}{2}\]

4. Circles

Curvilinear plane figures require proficiency in arc measure, sector area, and central vs. inscribed angle relationships.

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

5. 3D Solids

Mensuration 3D involves surface area and volume calculations for basic geometric solids.

Right Circular Cylinder

• Volume: \[V = \pi r^2 h\]
• Total Surface Area: \[A = 2\pi r h + 2\pi r^2\]

Right Circular Cone

• Volume: \[V = \frac{1}{3} \pi r^2 h\]
• Total Surface Area: \[A = \pi r s + \pi r^2 \quad (\text{where slant height } s = \sqrt{r^2 + h^2})\]

Sphere

• Volume: \[V = \frac{4}{3} \pi r^3\]
• Surface Area: \[A = 4 \pi r^2\]

6. Coordinate Geometry

Analytic geometry links algebraic equations to geometric figures on the Cartesian plane.

• Distance Formula: The distance \(d\) between points \((x_1, y_1)\) and \((x_2, y_2)\): \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
• Midpoint Formula: \[M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\]
• Slope (\(m\)): \[m = \frac{y_2 - y_1}{x_2 - x_1} = \tan \theta\]
• Equation of a Line (Slope-Intercept Form): \[y = mx + c\]
• Equation of a Circle: Centered at \((h, k)\) with radius \(r\): \[(x - h)^2 + (y - k)^2 = r^2\]

7. Transformations

Mapping points under basic geometric transformations in a coordinate space:

• Reflection across x-axis: \((x, y) \to (x, -y)\)
• Reflection across y-axis: \((x, y) \to (-x, y)\)
• Rotation \(90^{\circ}\) Counter-Clockwise (CCW): \((x, y) \to (-y, x)\)
• Dilation: With scale factor \(k\): \((x, y) \to (kx, ky)\)

People Also Ask (FAQ)

How many questions from Geometry appear in NDA Mathematics?

Direct questions on 2D/3D Geometry, Mensuration, and Analytical Geometry typically account for 18 to 25 questions in the NDA Mathematics paper.

Is Heron's formula essential for NDA geometry problems?

Yes, Heron's formula is frequently used in NDA questions involving scalene triangles where side lengths are known but the height is not explicitly given.

What is the relationship between inscribed and central angles in a circle?

The Inscribed Angle Theorem states that an angle subtended by an arc at the circumference is half of the angle subtended by the same arc at the center of the circle.