Geometry worked examples & tricks
In mensuration, the student who recognizes a pattern wins the time battle — and time, in competitive exams, is the real constraint.
Definition: Mensuration is the branch of mathematics that measures the lengths, areas, and volumes of geometric shapes and three-dimensional solids. Definition: CSA (Curved Surface Area) is the lateral surface area excluding any flat bases; TSA (Total Surface Area) includes all faces including the bases.
Formula toolkit
Commit these to memory — they are the building blocks of every problem:
| Shape | Area / CSA | TSA | Volume |
|---|---|---|---|
| Circle | πr² | — | — |
| Sector | (θ/360°) × πr² | — | — |
| Triangle | ½ × base × height | — | — |
| Rectangle | l × b | — | — |
| Cylinder | CSA = 2πrh | 2πr(r+h) | πr²h |
| Cone | CSA = πrl, l = √(r²+h²) | πr(r+l) | ⅓πr²h |
| Sphere | CSA = 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere | CSA = 2πr² | 3πr² | (2/3)πr³ |
| Cube | CSA = 4a² | 6a² | a³ |
| Cuboid | 2(lb+bh) | 2(lb+bh+hl) | l×b×h |
π convention: use 22/7 when numbers divide cleanly (radius = 7, 14, 21, etc.); otherwise use 3.14 or keep π in the answer.
Pythagorean triples — instant recognition saves minutes
Instead of using the Pythagorean theorem every time, recognize these common triples. A triple (a, b, c) satisfies a² + b² = c²:
| Triple | Multiples |
|---|---|
| 3 – 4 – 5 | 6-8-10, 9-12-15, 12-16-20 |
| 5 – 12 – 13 | 10-24-26 |
| 7 – 24 – 25 | 14-48-50 |
| 8 – 15 – 17 | 16-30-34 |
| 9 – 40 – 41 | — |
Worked examples
Question: A ladder of length 25 m leans against a wall with its foot 7 m from the wall. How high does it reach?
Solution:
Step 1: The sides form a right triangle with legs 7 and h, hypotenuse 25.
Step 2: Recognize the Pythagorean triple 7 – 24 – 25.
Conclusion: The ladder reaches 24 m (no calculation required once the triple is recognized).
Question: Find the volume of a cylinder of radius 7 m and height 10 m.
Solution:
Step 1: V = πr²h = (22/7) × 7² × 10.
Step 2: (22/7) × 49 × 10 = 22 × 7 × 10.
Conclusion: V = 1540 m³.
Question: A cone has base radius 6 cm and height 8 cm. Find its curved surface area.
Solution:
Step 1: Slant height l = √(r² + h²) = √(36 + 64) = √100 = 10 cm.
Step 2: CSA = πrl = π × 6 × 10.
Conclusion: CSA = 60π ≈ 188.4 cm².
Question: A sector has radius 14 cm and central angle 90°. Find its area.
Solution:
Step 1: Area of sector = (θ/360°) × πr² = (90/360) × (22/7) × 196.
Step 2: = (1/4) × 22 × 28 = (1/4) × 616.
Conclusion: Area = 154 cm².
Question: In a right triangle with legs 6 and 8, find the altitude drawn to the hypotenuse.
Solution:
Step 1: Hypotenuse = √(36 + 64) = 10 (triple 6-8-10).
Step 2: Altitude to hypotenuse = (product of legs) / hypotenuse = (6 × 8) / 10 = 48/10.
Conclusion: Altitude = 4.8 cm.
Question: A solid sphere of radius 6 cm is melted and recast as a cylinder of radius 0.2 cm. Find the cylinder's length.
Solution:
Step 1: Volume is conserved: V_sphere = V_cylinder.
Step 2: (4/3)π × 6³ = π × (0.2)² × L.
Step 3: Cancel π: (4/3) × 216 = 0.04 × L → 288 = 0.04 × L.
Conclusion: L = 288 / 0.04 = 7,200 cm = 72 m.
Question: Water flows through a pipe of diameter 2 cm at 3 m/s. Find the volume flow rate (in cm³/s).
Solution:
Step 1: Radius = 1 cm. Cross-sectional area = π × 1² = π cm².
Step 2: Speed = 300 cm/s (convert 3 m/s). Volume flow = area × speed = π × 300.
Conclusion: Flow rate ≈ 942.5 cm³/s.
The volume conservation principle
When a solid is melted, cut, or reshaped, total volume is conserved. This is the key to all "melted into" problems:
- Sphere melted into smaller spheres: (4/3)πR³ = n × (4/3)πr³ → R³ = n·r³ → n = (R/r)³.
- Cylinder melted into cones: πr₁²h₁ = n × (1/3)πr₂²h₂ → solve for n.
- Wire/rod: always πr²L = original volume.
Common traps and how to avoid them
Trap 1 — CSA vs TSA:
- A cylinder's CSA = 2πrh (only the curved wall).
- A cylinder's TSA = 2πrh + 2πr² = 2πr(r + h) (wall + two circular ends).
- If the question says "outer surface of a closed box," it wants TSA. An open cylinder (like a bucket) has TSA = CSA + one base = 2πrh + πr².
Trap 2 — Forgetting the square root for slant height:
Slant height l = √(r² + h²). Students often write l = r + h (arithmetic, not geometric). Always use the Pythagorean relation.
Trap 3 — Diameter vs radius:
When a problem gives diameter (say 14 cm), the radius is 7 cm. Using 14 in a formula for r gives a result off by a factor of 2 or 4.
Trap 4 — Units:
If radius is in m and height in cm, convert before substituting. Volume answer in m × cm² is dimensionally inconsistent.
Why it matters: Mensuration problems size physical objects — water tanks, grain silos, conical tents, pipes, bricks. The volume conservation idea behind the melting-sphere problem is the exact principle factories use to calculate how much wire a given mass of copper can yield, or how thick a sheet of metal will be after rolling.
Real-world example: A village overhead water tank shaped like a cylinder with radius 7 m and height 10 m holds 1,540 m³ of water — that is 15,40,000 litres. Using the same πr²h formula, the panchayat can determine whether the tank supplies enough for 500 families at 200 L per family per day (1,00,000 L/day — about 15 days' supply). This is exactly the cylinder-volume problem from the worked examples above.
Common misconception: Many candidates treat Pythagorean triples like 7–24–25 or 5–12–13 as things to derive from scratch using the formula every time. Memorizing all common triples (3-4-5, 5-12-13, 7-24-25, 8-15-17) lets you read off the third side instantly — that saves 20–30 seconds per problem, which is the difference between finishing a paper or not.
- ✓- Recognize Pythagorean triples (3-4-5, 5-12-13, 7-24-25, 8-15-17) to skip computation.
- ✓- Cone slant height l = √(r²+h²) — always use Pythagoras, never add r and h.
- ✓- CSA vs TSA: CSA excludes flat bases; TSA includes all surfaces.
- ✓- Cylinder volume = πr²h; cone volume = ⅓πr²h; sphere volume = (4/3)πr³.
- ✓- Melting/recasting problems: equate volumes of original and new solid.
- ✓- Sector area = (θ/360°) × πr²; altitude to hypotenuse = (product of legs)/hypotenuse.
"Cones are Third, Spheres are Four-Third" → V_cone = ⅓πr²h; V_sphere = (4/3)πr³. For slant: "l² = r² + h²" — Pythagoras, not addition.
- ✓- Pattern recognition (Pythagorean triples, common areas) beats brute-force calculation.
- ✓- Slant height of a cone always requires the Pythagorean relation l² = r² + h².
- ✓- Volume is always conserved when a solid is melted and recast — equate the two expressions.
- ✓- CSA and TSA are different; check whether the question wants the curved surface or the total.
- ✓- Use 22/7 when the radius is a multiple of 7; keep it as a fraction until the last step.
Triangles, circles & mensuration formulas
Geometry and mensuration questions are guaranteed marks in RRB, SSC, and most aptitude exams — if you have the formulas at your fingertips and know the shortcut triangles. This lesson compiles every essential result for triangles, polygons, circles, and 3-D solids, with the exact patterns examiners reuse year after year.
Definition: Mensuration is the branch of geometry concerned with measuring lengths, areas, and volumes of plane figures and solid bodies.
Definition: A Pythagorean triple is a set of three positive whole numbers (a, b, c) satisfying a² + b² = c², meaning they form the sides of a right-angled triangle.
Triangles — core properties and formulas
Every triangle obeys these four rules without exception:
- Angle sum: all three interior angles add to 180°; an exterior angle equals the sum of the two non-adjacent interior angles.
- Triangle inequality: each side must be less than the sum of the other two.
- Area = ½ × base × height. When the height is unknown, use Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a + b + c)/2 is the semi-perimeter.
- Equilateral triangle (side a): area = (√3/4)a², height = (√3/2)a.
Right triangle special cases:
The Pythagorean theorem a² + b² = c² (c = hypotenuse) is the backbone of half of all geometry questions. Memorise these triples to skip computation in the exam hall:
| Triple | Multiples to know |
|---|---|
| 3-4-5 | 6-8-10, 9-12-15, 15-20-25 |
| 5-12-13 | 10-24-26 |
| 8-15-17 | — |
| 7-24-25 | — |
| 20-21-29 | — |
Special angle ratios (derived from the unit circle):
- 30-60-90 triangle: sides in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).
- 45-45-90 triangle: sides in ratio 1 : 1 : √2 (equal legs : hypotenuse).
Why it matters: These ratios let you answer "find the diagonal of a square of side 7 cm" (= 7√2 ≈ 9.9 cm) in three seconds without any workings.
Quadrilaterals and polygons
| Shape | Area | Extra useful fact |
|---|---|---|
| Square (side a) | a² | diagonal = a√2 |
| Rectangle (l × b) | l × b | diagonal = √(l² + b²) |
| Parallelogram (base b, height h) | b × h | opposite sides equal |
| Rhombus (diagonals d₁, d₂) | ½ d₁ × d₂ | all sides equal; diagonals bisect perpendicularly |
| Trapezium (parallel sides a, b; height h) | ½(a + b) × h | only one pair of sides is parallel |
Regular n-gon: each interior angle = (n − 2) × 180° / n; sum of all interior angles = (n − 2) × 180°. For a regular hexagon (n = 6), each angle = 120°, and the side equals the circumradius — a fact repeatedly tested.
Circles — all the formulas you need
- Circumference = 2πr; Area = πr² (r = radius).
- Arc length of a sector = (θ/360°) × 2πr.
- Sector area = (θ/360°) × πr².
- Thales' theorem: the angle in a semicircle (i.e., the angle subtended by a diameter) = 90°.
- Tangent–radius: a tangent to a circle meets the radius at the point of contact at 90°.
Common misconception: Students often confuse arc length with chord length. The arc is the curved portion of the circle's circumference; the chord is the straight line joining the two endpoints. They differ unless the angle is 180°.
3-D mensuration
| Solid | Volume | Lateral / Curved SA | Total SA |
|---|---|---|---|
| Cube (side a) | a³ | 4a² | 6a² |
| Cuboid (l, b, h) | lbh | 2h(l + b) | 2(lb + bh + hl) |
| Cylinder (r, h) | πr²h | 2πrh | 2πr(r + h) |
| Cone (r, h, slant l) | ⅓πr²h | πrl | πr(r + l) |
| Sphere (r) | (4/3)πr³ | — | 4πr² |
| Hemisphere (r) | (2/3)πr³ | 2πr² | 3πr² |
For a cone, slant height l = √(r² + h²). This is the Pythagorean theorem applied to the right triangle formed by the radius, the vertical height, and the slant side. Using h instead of l in πrl is the single most common mensuration error in exams.
Space diagonals: cube → a√3; cuboid → √(l² + b² + h²). These come up frequently in questions about the longest rod that can fit inside a box.
Worked example — recognising a Pythagorean triple
Question: A ladder's foot is 5 m from a wall and its top reaches 12 m up. How long is the ladder?
Solution:
Step 1: The wall, the ground, and the ladder form a right triangle with legs 5 m and 12 m.
Step 2: Recognise the 5-12-13 Pythagorean triple instead of squaring and taking a square root.
Step 3: The hypotenuse is therefore 13 m.
Conclusion: The ladder is 13 m long.
Worked example — cone vs cylinder
Question: A cone and a cylinder have the same base radius r = 7 cm and the same height h = 24 cm. Find the slant height of the cone and compare the volumes.
Solution:
Step 1: Slant height l = √(r² + h²) = √(49 + 576) = √625 = 25 cm (recognise 7-24-25 triple!).
Step 2: V_cylinder = πr²h = π × 49 × 24 = 1176π cm³.
Step 3: V_cone = ⅓ × 1176π = 392π cm³.
Conclusion: The cone's volume is exactly one-third of the cylinder's — a universal ratio when base and height match.
Real-world example: An Indian painter is asked to paint a cylindrical overhead water tank (r = 2.1 m, h = 4 m). He needs the curved surface area = 2πrh = 2 × 22/7 × 2.1 × 4 = 52.8 m² of paint. He uses TSA = 2πr(r + h) if the top lid is also painted. Confusing the two gives him either too little or too much paint — a direct cost error.
- ✓- Triangle angles sum to 180°; Heron's formula handles any triangle when only sides are known.
- ✓- Memorise triples 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and special ratios 1:√3:2 (30-60-90) and 1:1:√2 (45-45-90).
- ✓- Circle: circumference 2πr, area πr²; angle in semicircle = 90°; tangent ⊥ radius.
- ✓- Arc length and sector area both use the same (θ/360°) fraction.
- ✓- Cone CSA = πrl where l = √(r² + h²), not h.
- ✓- Sphere: V = (4/3)πr³, SA = 4πr²; hemisphere TSA = 3πr².
- ✓- Regular n-gon interior angle = (n − 2) × 180° / n.
- ✓- V_cone = ⅓ V_cylinder when base radius and height are the same.
"CSL" for cone: Curved surface = π × r × L (slant), not height.
Slant height is the Longest side of the right triangle (r, h, l) — the hypotenuse.
- ✓- Keep all area and volume formulas memorised for guaranteed exam marks.
- ✓- Spot Pythagorean triples in disguise to skip square-root computation entirely.
- ✓- Always distinguish a cone's vertical height from its slant height.
- ✓- Use the θ/360° multiplier for any partial-circle (arc/sector) question.
- ✓- The diagonal of a cube is a√3; of a cuboid is √(l² + b² + h²).
- ✓- V_cone : V_cylinder = 1 : 3 for same base and height — a universal ratio.
Geometry & Mensuration — Core
Geometry & Mensuration is a high-yield, formula-driven block of the RRB Group D Maths section (25 questions total). Expect 2–4 direct-substitution questions here — pure marks if your formulas are memorised. Nothing needs proof; you just plug numbers, usually with π = 22/7.
Must-know formulas
| Shape | Area / Volume | Perimeter / Surface |
|---|---|---|
| Square (side a) | a² | 4a; diagonal a√2 |
| Rectangle (l,b) | l×b | 2(l+b); diagonal √(l²+b²) |
| Triangle | ½ × base × height | — |
| Equilateral (a) | (√3/4)a² | height (√3/2)a |
| Circle (r) | πr² | 2πr |
| Cube (a) | a³ | TSA 6a²; diagonal a√3 |
| Cuboid (l,b,h) | l·b·h | TSA 2(lb+bh+hl) |
| Cylinder (r,h) | πr²h | CSA 2πrh; TSA 2πr(r+h) |
| Cone (r,h) | ⅓πr²h | CSA πrl, l=√(r²+h²) |
| Sphere (r) | (4/3)πr³ | 4πr² |
Exam Tricks & Tips
- 🎯 Take r = 7 or 14 friendly: with π=22/7 the 7 cancels, e.g. circle r=7 → area 154, circumference 44.
- 🎯 Memorise Pythagoras triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 — they instantly give diagonals/hypotenuses.
- 🎯 Scaling rule: if every side is doubled, area becomes 4×, volume becomes 8× — answer without recomputing.
- 🎯 Equilateral-triangle area = (√3/4)a²; keep √3 ≈ 1.732 for decimal options.
- 🎯 Cube/cuboid space diagonal = √(l²+b²+h²); for a cube it is a√3.
- 🎯 Cylinder vs cone same r,h: cone volume = ⅓ of cylinder — many questions test exactly this.
- ❌ Common mistake: mixing CSA and TSA — CSA leaves out the flat circular ends; TSA adds them.
Expected exam pattern
"Area of a circle of radius 7 cm", "Volume of a cube of side 6 cm", or a two-step "cost of fencing/painting" wrapper around perimeter or surface area. All single-step substitution.
Quick recap
Learn the table cold, keep π=22/7 and √3≈1.732 handy, use radius 7/14 shortcuts, and never confuse CSA with TSA.
Geometry & Mensuration — Core
Cover the answer, recall, then check. 12 must-know mensuration facts for RRB Group D (use π = 22/7).
Q1. Area of a circle of radius 7 cm?
A1. πr² = 22/7 × 49 = 154 cm².
Q2. Circumference of a circle of radius 7 cm?
A2. 2πr = 2 × 22/7 × 7 = 44 cm.
Q3. Volume of a cube of side 5 cm? Its total surface area?
A3. Volume a³ = 125 cm³; TSA 6a² = 6×25 = 150 cm².
Q4. Diagonal of a cube of side 4 cm?
A4. a√3 = 4√3 ≈ 6.93 cm.
Q5. Area of an equilateral triangle of side 4 cm?
A5. (√3/4)a² = (√3/4)×16 = 4√3 ≈ 6.93 cm².
Q6. Volume of a cylinder, r = 7 cm, h = 10 cm?
A6. πr²h = 22/7 × 49 × 10 = 1540 cm³.
Q7. TSA of a cuboid 5×4×3 cm?
A7. 2(lb+bh+hl) = 2(20+12+15) = 94 cm².
Q8. Perimeter of a square whose area is 64 cm²?
A8. Side = 8, perimeter 4a = 32 cm.
Q9. Diagonal of a rectangle 3 cm × 4 cm?
A9. √(9+16) = 5 cm (3-4-5 triple).
Q10. Volume of a cone, r = 7 cm, h = 6 cm?
A10. ⅓πr²h = ⅓ × 22/7 × 49 × 6 = 308 cm³.
Q11. Volume of a sphere of radius 3 cm?
A11. (4/3)πr³ = 4/3 × 22/7 × 27 = ≈ 113.1 cm³.
Q12. Area of a trapezium, parallel sides 8 and 12 cm, height 5 cm?
A12. ½(a+b)h = ½ × 20 × 5 = 50 cm².
Geometry & Mensuration — Core
RRB Group D combines a few triangle facts with the area and perimeter formulas of everyday shapes. Learn a short formula list and label the figure carefully, and these are quick, reliable marks.
The core idea
Geometry describes shapes and angles; mensuration measures them — perimeter (boundary length), area (surface covered), and, for solids, volume.
Beginner — triangle facts
- Angles of a triangle sum to 180°.
- Right triangle: a² + b² = c² (Pythagoras), c the hypotenuse.
- Perimeter = sum of the three sides.
Intermediate — areas and perimeters of flat shapes
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | length × breadth | 2(length + breadth) |
| Square | side² | 4 × side |
| Triangle | ½ × base × height | sum of sides |
| Circle | π × r² | 2 × π × r |
Take π ≈ 22/7 unless told otherwise.
Advanced — combined figures
For an L-shape or a shape with a semicircular end, split it into a rectangle plus a triangle or half-circle, find each area, and add.
Worked example
A rectangular field is 40 m long and 30 m wide. Find its area and the length of fencing needed to enclose it.
Area = length × breadth = 40 × 30 = 1,200 m².
Perimeter (fencing) = 2 × (40 + 30) = 2 × 70 = 140 m.
Exam relevance
Group D asks for the area or perimeter of a rectangle, square, triangle, or circle, or a simple Pythagoras length. Direct formula use wins.
Tricks and mnemonic
Perimeter is a length (metres); area is a surface (square metres). If your answer's unit does not match what is asked, you used the wrong formula. For circles chant "area is pi-r-squared, boundary is two-pi-r."
Do not use the slant side as the height of a triangle. The height must be perpendicular to the base you chose.
- ✓- Triangle angles sum to 180°; Pythagoras gives right-triangle sides.
- ✓- Rectangle area = l × b; perimeter = 2(l + b).
- ✓- Circle area = πr²; circumference = 2πr; use π ≈ 22/7.
- ✓- Triangle area = ½ × base × perpendicular height.
- ✓- Split compound shapes into simple pieces and add.
- ✓- Perimeter is boundary length; area is surface covered.
- ✓- Memorise the four core area and perimeter formulas.
- ✓- Height in a triangle is always perpendicular to the base.
- ✓- Check that your unit (m vs m²) matches the question.