Geometry and Mensuration
Geometry in GATE’s General Aptitude section tests your ability to visualize shapes, apply area/volume formulas, and use properties of similar figures. Mensuration (3D geometry) is a perennial favorite — the formulas are fixed, and so are the ways GATE tricks you.
🟢 Lite — Quick Review (1h–1d)
Must-know formulas:
- Triangle: Area = ½ × base × height; Perimeter = a + b + c
- Circle: Area = πr²; Circumference = 2πr
- Rectangle: Area = l × w; Perimeter = 2(l + w)
- Cube: Volume = a³; Surface area = 6a²
- Cylinder: Volume = πr²h; Curved SA = 2πrh; Total SA = 2πr(r + h)
- Sphere: Volume = (4/3)πr³; Surface area = 4πr²
- Pythagorean theorem: a² + b² = c² (for right-angled triangles)
⚡ GATE exam tip: When a problem mentions “the area of a semicircle,” the formula is (1/2)πr² — don’t forget the π. GATE often tests whether you retain π or approximate it as 22/7.
⚡ Quick trick: In an isosceles triangle with equal sides a and base b, the height = √(a² − b²/4). Use this when asked for area or altitude.
⚡ Common trap: Don’t confuse curved surface area (lateral surface) with total surface area. For a cone, CSA = πrl, TSA = πr(r + l).
🟡 Standard — Regular Study (2d–2mo)
Triangles
Classification by sides:
- Equilateral: all sides equal, all angles = 60°
- Isosceles: two sides equal
- Scalene: all sides different
Area formulas:
- ½ × base × height
- Heron’s formula: √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2 (semiperimeter)
- For equilateral triangle of side a: Area = (√3/4)a²
Right-angled triangle (Pythagorean triples):
| Set | Ratio |
|---|---|
| 3-4-5 | 3:4:5 |
| 5-12-13 | 5:12:13 |
| 7-24-25 | 7:24:25 |
| 8-15-17 | 8:15:17 |
GATE Example (2019, 1 mark): Find the area of a triangle with sides 9, 10, and 17 cm.
s = (9+10+17)/2 = 18. Area = √[18×9×8×1] = √1296 = 36 cm². (Note: 9²+10²=181 ≠ 289=17², so not right-angled.)
Similar triangles: If two triangles are similar, the ratio of their areas = (ratio of corresponding sides)².
Circles
- Area: πr²
- Circumference: 2πr
- Arc length: (θ/360) × 2πr (where θ is the central angle in degrees)
- Sector area: (θ/360) × πr²
GATE Example: A circle has radius 7 cm. Find the area of a sector with a 60° angle.
Area = (60/360) × π × 7² = (1/6) × 22/7 × 49 = 77/3 ≈ 25.67 cm².
Polygons
Regular n-gon:
- Each interior angle = [(n−2) × 180°] / n
- Each exterior angle = 360°/n
- Sum of interior angles = (n−2) × 180°
Key polygons to know:
| Polygon | Number of sides | Sum of angles |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
3D Mensuration — Key Solids
Cube (side a):
- Volume = a³
- Surface area = 6a²
- Diagonal = a√3
Cuboid (l × b × h):
- Volume = l × b × h
- Surface area = 2(lb + bh + hl)
- Diagonal = √(l² + b² + h²)
Cylinder (radius r, height h):
- Volume = πr²h
- Curved surface area = 2πrh
- Total surface area = 2πr(r + h)
Cone (radius r, height h, slant height l):
- l = √(r² + h²)
- Volume = (1/3)πr²h
- Curved surface area = πrl
- Total surface area = πr(r + l)
Sphere (radius r):
- Volume = (4/3)πr³
- Surface area = 4πr²
GATE Example (2017, 1 mark): The total surface area of a cube is 216 cm². Find its volume.
6a² = 216 → a² = 36 → a = 6 cm. Volume = 6³ = 216 cm³.
🔴 Extended — Deep Study (3mo+)
Similar Figures — Area and Volume Ratios
Important principle: When two similar figures have corresponding sides in ratio k:1, then:
- Their areas are in ratio k²:1
- Their volumes are in ratio k³:1
GATE Advanced Example: Two spheres have radii in the ratio 2:3. Find the ratio of their volumes.
Volume ratio = (2³):(3³) = 8:27.
Prism and Pyramid
Right Prism:
- Volume = Base area × height
- Lateral surface area = Perimeter of base × height
Regular Pyramid:
- Volume = (1/3) × Base area × height
- Lateral surface area = (1/2) × Perimeter of base × slant height
Frustum of a Cone
When a cone is sliced parallel to its base, the remaining portion is a frustum:
- Volume = (1/3)πh(R² + r² + Rr) where R and r are the radii of the two bases
- CSA = π(R + r) × slant height (where slant height = √[h² + (R−r)²])
Hemisphere
Hemisphere (radius r):
- Volume = (2/3)πr³
- Curved surface area = 2πr²
- Total surface area = 3πr² (including the base circle)
Combination of Solids
When multiple solids are combined (e.g., a cone on top of a cylinder), add volumes and surface areas carefully — shared faces are NOT exposed in the total surface area.
GATE Example: A wooden article is made by scooping out a hemisphere from each end of a solid cylinder of height 14 cm and radius 3 cm. Find the remaining volume.
Cylinder volume = π×3²×14 = 126π. Two hemispheres = 1 full sphere = (4/3)π×3³ = 36π. Remaining = 126π − 36π = 90π cm³.
Coordinate Geometry Basics
Distance between two points (x₁, y₁) and (x₂, y₂):
d = √[(x₂−x₁)² + (y₂−y₁)²]
Section formula (internal division):
Point dividing line joining (x₁, y₁) and (x₂, y₂) in ratio m:n = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃):
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Pythagorean Theorem Applications
GATE trick: The converse of Pythagorean theorem also works: if a² + b² = c², the triangle is right-angled with c as the hypotenuse.
Distance from a point to a line:
Distance from (x₀, y₀) to Ax + By + C = 0 is |Ax₀ + By₀ + C| / √(A² + B²)
Angle Bisector Theorem
In triangle ABC, if AD bisects angle A (meeting BC at D):
BD/DC = AB/AC
Quadrilaterals
| Type | Properties |
|---|---|
| Parallelogram | Opposite sides parallel, opposite angles equal, diagonals bisect each other |
| Rectangle | Parallelogram + right angles, diagonals equal |
| Rhombus | Parallelogram + all sides equal, diagonals perpendicular |
| Square | Rectangle + Rhombus |
| Trapezium | One pair of parallel sides |
Area of trapezium = ½(sum of parallel sides) × height
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