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Numerical Ability · AFCAT

NA08 — Geometry & Mensuration (2D)

📐 Numerical Ability – NA08 AFCAT Level ☆ Low Priority
📌 AFCAT Focus 2022–2026: 1–2 questions, usually on area/perimeter of triangles, circles, and rectangles. Right-triangle Pythagoras and circle area are the most tested. Memorise the formula table — most AFCAT geometry is formula look-up plus one substitution.

1. Triangles

Fig. 1.1 — Triangle Types, Properties & Area Formulas
TRIANGLES — Types, Properties & Key Formulas EQUILATERAL All sides = a · All angles = 60° Area = (√3/4) a² Perimeter = 3a Height = (√3/2) a Inradius = a / (2√3) All angles = 60° RIGHT-ANGLED b a (base) hyp h² = a² + b² Area = ½ × base × height Pythagorean Triples: 3–4–5 · 5–12–13 8–15–17 · 7–24–25 GENERAL / SCALENE Sides a, b, c (all different) Area = ½ × b × h Heron's Formula: √s(s−a)(s−b)(s−c) s = (a + b + c) / 2 Perimeter = a + b + c
TypesAll Triangle Types & Key PropertiesAFCAT Direct
Equilateral
All 3 sides equal. All 3 angles = 60°. Area = (√3/4)a². Height = (√3/2)a.
Isosceles
2 sides equal → angles opposite equal sides are also equal. Area = (b/4)√(4a²−b²) where b = base, a = equal side.
Scalene
All 3 sides and all 3 angles different. Use Heron's formula: Area = √s(s−a)(s−b)(s−c), s = (a+b+c)/2.
Right-angled
One angle = 90°. Hypotenuse² = base² + height² (Pythagoras theorem). Key triples: 3–4–5, 5–12–13, 8–15–17, 7–24–25.
Angle Sum
Sum of interior angles of ANY triangle = 180°. Exterior angle = sum of two non-adjacent interior angles.
Triangle Inequality
For a valid triangle: sum of any two sides > third side. i.e. a+b > c, b+c > a, a+c > b. Fails = no triangle possible.
Centroid / Median
Median joins vertex to midpoint of opposite side. Centroid divides each median in 2:1 ratio from vertex. 3 medians meet at centroid.

2. Quadrilaterals & Circles

Fig. 2.1 — Quadrilaterals & Circle: Shapes, Formulas & Key Properties
QUADRILATERALS & CIRCLE — Area, Perimeter & Key Facts SHAPE AREA & PERIMETER KEY FACT Square side = a Area = a² Perimeter = 4a Diagonal = a√2 Rectangle length l, breadth b Area = l × b Perimeter = 2(l + b) Diagonal = √(l²+b²) Parallelogram base b, height h Area = b × h Perimeter = 2(a + b) Opp sides = & ∥ Rhombus diagonals d₁, d₂ Area = ½ d₁ × d₂ Perimeter = 4a Diagonals bisect at 90° Circle (r) Area = πr²  |  Circumference = 2πr Semi area = πr²/2
CIRCLES & POLYGONS

3. Circle Properties & Regular Polygons

⭕ Circle — Full Reference

  • Radius (r): Distance from centre to any point on circle
  • Diameter (d) = 2r: Longest chord; passes through centre
  • Circumference = 2πr = πd
  • Area = πr²
  • Semicircle: Area = πr²/2 · Perimeter = πr + 2r
  • Quarter circle: Area = πr²/4 · Perimeter = πr/2 + 2r
  • Sector (angle θ): Area = (θ/360)×πr² · Arc length = (θ/360)×2πr
  • Chord: Any line segment with both ends on circle
  • Tangent: Touches circle at exactly one point; perpendicular to radius at that point

🔷 Regular Polygons

  • Sum of interior angles of any polygon = (n−2) × 180°
  • Each interior angle (regular polygon) = (n−2)×180° / n
  • Each exterior angle = 360° / n
  • Pentagon (n=5): Sum = 540°. Each angle = 108°. Area = (5s²/4)×cot(36°) ≈ 1.72s²
  • Hexagon (n=6): Sum = 720°. Each angle = 120°. Area = (3√3/2)s²
  • Octagon (n=8): Sum = 1080°. Each angle = 135°
  • Number of diagonals = n(n−3)/2
  • Pentagon diagonals = 5. Hexagon diagonals = 9
ShapeSides (n)Sum of AnglesEach Interior AngleKey Area Formula
Triangle3180°60° (equilateral)½ × base × height
Quadrilateral4360°90° (square/rect)varies by shape
Pentagon5540°108°≈ 1.72 × s²
Hexagon6720°120°(3√3/2) × s²
Octagon81080°135°2(√2+1) × s²

📐 Formula Sheet — NA08

Triangle Core
Area = ½ × base × height
Equilateral: (√3/4)a²
Heron's: √s(s−a)(s−b)(s−c)
s = (a+b+c)/2
Pythagoras Triples
3–4–5 (×2: 6–8–10)
5–12–13
8–15–17
7–24–25
Quadrilaterals
Square: A = a² | P = 4a | diag = a√2
Rectangle: A = lb | P = 2(l+b)
Parallelogram: A = b×h
Trapezium: A = ½(a+b)×h
Circle
Area = πr²
Circumference = 2πr
Semicircle area = πr²/2
Sector area = (θ/360)×πr²
Angle Properties
Triangle angles sum = 180°
Polygon angles sum = (n−2)×180°
Each angle (regular n-gon) = (n−2)×180°/n
Exterior angle = 360°/n
Rhombus
Area = ½ × d₁ × d₂ (diagonals)
Perimeter = 4a (all sides equal)
Diagonals bisect at 90°
Side = √((d₁/2)²+(d₂/2)²)

📝 Topic-Wise PYQs — NA08

Q1. The area of an equilateral triangle with side 6 cm is: AFCAT PYQ
(a) 9√3 cm²(b) 18 cm²(c) 6√3 cm²(d) 36 cm²
✔ Answer: (a) 9√3 cm²
Area = (√3/4)×6² = (√3/4)×36 = 9√3 cm². Remember: for equilateral triangle always use (√3/4)a².
Q2. A rectangle has length 12 m and diagonal 13 m. Find its area. AFCAT PYQ
(a) 48 m²(b) 60 m²(c) 65 m²(d) 72 m²
✔ Answer: (b) 60 m²
Breadth = √(13²−12²) = √(169−144) = √25 = 5 m. Area = 12×5 = 60 m². (5-12-13 is a Pythagorean triple.)
Q3. The circumference of a circle is 44 cm. Find its area. (π = 22/7) AFCAT PYQ
(a) 154 cm²(b) 176 cm²(c) 144 cm²(d) 196 cm²
✔ Answer: (a) 154 cm²
2πr = 44 → r = 44/(2×22/7) = 44×7/44 = 7 cm. Area = πr² = (22/7)×49 = 154 cm².
Q4. The parallel sides of a trapezium are 10 cm and 6 cm, height 4 cm. Find area. AFCAT PYQ
(a) 28 cm²(b) 32 cm²(c) 40 cm²(d) 60 cm²
✔ Answer: (b) 32 cm²
Area = ½(a+b)×h = ½(10+6)×4 = ½×16×4 = 32 cm².

🧠 Quick Memory Chart — NA08

📐 Triangles
  • General: ½ × b × h
  • Equilateral: (√3/4)a²
  • Heron: √s(s-a)(s-b)(s-c)
  • Pyth triples: 3-4-5, 5-12-13
📐 Quads
  • Square: a² | 4a
  • Rect: lb | 2(l+b)
  • Rhombus: ½d₁d₂
  • Trapezium: ½(a+b)h
📐 Circle
  • Area: πr²
  • Circumf: 2πr
  • Sector: (θ/360)πr²
  • Semi-area: πr²/2
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Q5. The sum of interior angles of a hexagon is: AFCAT PYQ
(a) 540°(b) 720°(c) 900°(d) 1080°
✔ Answer: (b) 720°
Sum = (n−2)×180° = (6−2)×180° = 4×180° = 720°. Each interior angle of a regular hexagon = 720°/6 = 120°.
Q6. The area of a rhombus is 120 cm². If one diagonal is 24 cm, find the other diagonal. AFCAT PYQ
(a) 8 cm(b) 10 cm(c) 12 cm(d) 15 cm
✔ Answer: (b) 10 cm
Area = ½ × d₁ × d₂ → 120 = ½ × 24 × d₂ → d₂ = 240/24 = 10 cm.