Mathematics

Geometry & Mensuration — 2D and 3D Shapes

Geometry tests your understanding of lines, angles, triangles and circles, while Mensuration tests area, perimeter, volume and surface area of shapes. Together they contribute 4–6 marks in SSC, 3–4 in RRB Group D and 2–3 in Banking exams.

Exam relevance: SSC CGL Tier-I and Tier-II both contain pure geometry questions, often with options that look similar. Mensuration is heavily tested in RRB and state PSC exams. Knowing the standard formulas saves precious seconds.

On this page

  1. 2D Shapes — Area & Perimeter
  2. 3D Shapes — Volume & Surface Area
  3. Basic Geometry — Lines, Angles, Triangles

12D Shapes — Area & Perimeter

Area is the surface enclosed by a 2D figure; perimeter is the total length of its boundary.

Square: Area = a², Perimeter = 4a, Diagonal = a√2. Rectangle: Area = l×b, Perimeter = 2(l+b), Diagonal = √(l²+b²). Triangle: Area = ½ × base × height; for sides a, b, c use Heron's formula = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. Equilateral triangle of side a: Area = (√3/4)a²; Height = (√3/2)a. Circle of radius r: Area = πr², Circumference = 2πr. Parallelogram: Area = base × height. Trapezium: Area = ½(a+b) × h.

Examples
  • Square with side 7 cm → Area = 49 cm², Perimeter = 28 cm, Diagonal = 7√2 ≈ 9.9 cm.
  • Rectangle 12×5 → Area = 60, Perimeter = 34, Diagonal = √169 = 13.
  • Equilateral triangle of side 6 → Area = (√3/4) × 36 = 9√3 ≈ 15.59 cm².
  • Circle r = 7 → Area = (22/7)×49 = 154 cm², Circumference = 2×(22/7)×7 = 44 cm.
Exam tip: If a square and a circle have the same perimeter, the circle has the larger area. If they have the same area, the circle has the smaller perimeter.

23D Shapes — Volume & Surface Area

Volume measures the space a 3D solid occupies; surface area is the total area of all its faces.

Cube (side a): Volume = a³, Total Surface Area (TSA) = 6a², Diagonal = a√3. Cuboid (l×b×h): V = lbh, TSA = 2(lb+bh+hl), Diagonal = √(l²+b²+h²). Cylinder (radius r, height h): V = πr²h, Curved SA = 2πrh, TSA = 2πr(r+h). Cone (radius r, height h, slant l = √(r²+h²)): V = (1/3)πr²h, Curved SA = πrl, TSA = πr(r+l). Sphere (radius r): V = (4/3)πr³, SA = 4πr². Hemisphere: V = (2/3)πr³, Curved SA = 2πr², TSA = 3πr².

Examples
  • Cube of side 5 cm → V = 125 cm³, TSA = 150 cm², Diagonal = 5√3.
  • Cylinder r = 7, h = 10 → V = (22/7)×49×10 = 1540 cm³.
  • Cone r = 6, h = 8 → slant l = 10; CSA = π×6×10 = 60π cm².
  • Sphere r = 7 → V = (4/3)×(22/7)×343 ≈ 1437.33 cm³; SA = 4×(22/7)×49 = 616 cm².
Exam tip: When a sphere of radius r is melted into a cylinder/cone, equate volumes — never surface areas.

3Basic Geometry — Lines, Angles, Triangles

A line extends infinitely in both directions; an angle is formed by two rays meeting at a point. A triangle is a closed three-sided polygon.

Sum of angles on a straight line = 180°; around a point = 360°; in a triangle = 180°; in a quadrilateral = 360°; in any n-gon = (n−2)×180°. Triangle types: by sides — Scalene, Isosceles, Equilateral; by angles — Acute, Right, Obtuse. Pythagoras theorem: in a right triangle, hypotenuse² = base² + height². Similar triangles have equal angles and proportional sides; the ratio of areas of similar triangles = (ratio of corresponding sides)².

Examples
  • Two angles of a triangle are 40° and 75° → third angle = 180 − 115 = 65°.
  • Right triangle with legs 9 and 12 → hypotenuse = √(81+144) = 15.
  • Sum of interior angles of a hexagon = (6−2) × 180 = 720°.
  • Two triangles are similar with sides in ratio 2:3 → ratio of areas = 4:9.
Exam tip: In a 30°-60°-90° triangle, sides are in ratio 1 : √3 : 2. In a 45°-45°-90°, sides are in ratio 1 : 1 : √2. Memorise these.

Short Tricks & Shortcuts

Use these speed tricks in the exam. Each trick is followed by a worked example so you can verify the shortcut yourself.

Trick 1Pythagorean triplets — memorise the primitives

Memorise: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29. Any multiple (e.g. 6-8-10, 9-12-15) is also a triplet.

Example: Right triangle, legs 9 and 12 → hypotenuse must be 15 (3-4-5 × 3).
Trick 2Equilateral triangle quick formulas

For side a: Area = (√3/4)a², Height = (√3/2)a, Inradius = a/(2√3), Circumradius = a/√3.

Example: Side 6 → Area = 9√3 ≈ 15.59 cm², Height = 3√3 ≈ 5.2 cm.
Trick 3Same perimeter — circle has the largest area

Among shapes with equal perimeter, circle > square > rectangle in area. Among shapes with equal area, circle has the smallest perimeter.

Example: Wire of 88 cm bent into a circle: r=14, area = 616 cm². Same wire as a square of side 22: area = 484 cm². Circle wins.
Trick 4Diagonals in cubes and cuboids

Cube of side a: diagonal = a√3. Cuboid l×b×h: diagonal = √(l² + b² + h²). Square of side a: diagonal = a√2.

Example: Cuboid 3×4×12 → diagonal = √(9+16+144) = √169 = 13.
Trick 5Sum of interior angles of a polygon

For an n-sided polygon: sum of interior angles = (n − 2) × 180°. Each interior angle of a regular polygon = (n−2)×180°/n.

Example: Hexagon (n=6) → sum = 720°; each angle of regular hexagon = 720/6 = 120°.
Trick 630-60-90 and 45-45-90 ratios

30-60-90 triangle sides are in ratio 1 : √3 : 2. 45-45-90 sides are in ratio 1 : 1 : √2. Always identify the triangle type first.

Example: Right triangle with hypotenuse 10, one angle 30° → other sides = 5 and 5√3.
Trick 7Volume of melted sphere into cylinder

When solids are melted/recast, equate VOLUMES (not surface areas). (4/3)πr³ = πR²H gives the new height/radius.

Example: Sphere r=6 cm melted into cylinder of base radius 4 cm → height = (4/3 × 216)/16 = 18 cm.

Quick Revision Facts

  • Diagonal of a cube of side a = a√3; of a square of side a = a√2.
  • Volumes of similar solids are in the cube of the side ratio.
  • Use π = 22/7 when the radius is a multiple of 7; otherwise use 3.14.
  • Angles in the same segment of a circle are equal; angle subtended by a diameter is 90°.

Frequently Asked Questions

Use ½ × base × height when you know one side and the perpendicular height. Use Heron's formula when you only know the three sides — it requires computing s = (a+b+c)/2 first.

Memorise the small primitive triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29. Any multiple (e.g. 6-8-10) is also a Pythagorean triplet.
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