Geometry & Mensuration

Geometry & Mensuration

🟢 Lite

Key Formula/Rule

Geometry and mensuration test your ability to calculate perimeters, areas, and volumes of 2D and 3D shapes. Master the core formulas and know when to use each one.

Memory Trick

“A-Rect = LB” — Area of a Rectangle is Length × Breadth. For a circle, “Pie Are Squared” — πr² (pi × r squared, not “pie are square”!). For volume, imagine filling a box with unit cubes.

1-Sentence Summary

Geometry requires memorising area and perimeter formulas for 2D shapes, and surface area and volume for 3D shapes — plus knowing when to use π (curved surfaces) versus when not to.

30-Second Formula Reference

Shape	Area	Perimeter/Circumference
Rectangle	l × b	2(l + b)
Square	a²	4a
Triangle	½ × base × height	Sum of 3 sides
Circle	πr²	2πr
Parallelogram	base × height	2(a + b)
Trapezium	½(a + b) × height	Sum of parallel sides + legs

Quick Examples

Q1: Area of a circle with radius 7 cm. A: π × 7² = 22/7 × 49 = 154 cm²

Q2: Perimeter of a rectangle 12 cm × 8 cm. A: 2(12 + 8) = 40 cm

Q3: Surface area of a cube with side 5 cm. A: 6a² = 6 × 25 = 150 cm²

Must Remember — 3D Shapes

Shape	TSA	Volume
Cube	6a²	a³
Cuboid	2(lb + bh + hl)	l × b × h
Cylinder	2πrh + 2πr²	πr²h
Cone	πrℓ + πr²	⅓πr²h
Sphere	4πr²	⁴⁄₃πr³
Hemisphere	3πr²	⅔πr³

Note: For cylinder, “total surface area” includes both circles. “Curved surface area” (CSA) = 2πrh only.

Exam Tips for CUET

• CUET often combines shapes — e.g., a tent (cone on cylinder). Break it into parts.
• Always check whether the question asks for curved surface area (CSA) or total surface area (TSA).
• π ≈ 22/7 or 3.14 unless specified otherwise — use the value that gives clean numbers.
• Unit consistency: If side is in cm, area is in cm². Convert all dimensions to the same unit before calculating.
• For composite shapes, calculate each component separately and add or subtract as required.

Common Pitfalls

• Using diameter instead of radius: Area = πr², NOT πd². The diameter is 2r — double-check.
• Forgetting that trapezium area uses the AVERAGE of parallel sides: ½(a + b) × h, not a × b × h.
• Mixing up volume and surface area units: Volume is cm³ (cubed), surface area is cm² (squared).
• Not simplifying compound shapes: A shape with a semicircle attached to a rectangle — calculate each separately.

🟡 Standard

Concept

Geometry & Mensuration is all about measuring shapes — how much space they take up (area), how long their edges are (perimeter/circumference), and how much stuff fits inside them (volume). You’ve been dealing with these since middle school, so you’re not starting from zero here.

2D shapes have two dimensions: length and breadth (or just one side for squares). Their area is the space inside, and their perimeter is the total distance around the boundary. Common ones you’ll see in CUET: triangles, rectangles, squares, circles, parallelograms, trapeziums, and rhombuses.

3D shapes add a third dimension: height or depth. Now you’re dealing with volume (how much space inside) and also surface area — the total area of all the outer faces. Cubes, cuboids, cylinders, cones, spheres, and hemispheres are the usual suspects.

The trickiest part for most students isn’t the formulas — it’s knowing which formula to apply and making sure your units are consistent (no mixing cm² with m²!).

Key Formulas

Formula	Use
Square: Area = a², Perimeter = 4a	When all 4 sides are equal
Rectangle: Area = l × b, Perimeter = 2(l + b)	When opposite sides are equal
Triangle: Area = ½ × base × height	Any triangle
Circle: Area = πr², Circumference = 2πr	Round shapes
Parallelogram: Area = base × height	Opposite sides parallel
Rhombus: Area = ½ × d₁ × d₂	Diagonal-based formula
Trapezium: Area = ½ × (a + b) × h	One pair of parallel sides
Cube: Volume = a³, TSA = 6a²	All edges equal
Cuboid: Volume = l × b × h, TSA = 2(lb + bh + hl)	Rectangular box
Cylinder: Volume = πr²h, CSA = 2πrh, TSA = 2πr(r + h)	Tube-shaped
Cone: Volume = ⅓πr²h, CSA = πrl, TSA = πr(r + l)	Ice-cream cone shape
Sphere: Volume = ⁴⁄₃πr³, TSA = 4πr²	Ball-shaped
Hemisphere: Volume = ⅔πr³, CSA = 2πr², TSA = 3πr²	Half a sphere

Worked Example

Q: A hall is 20 m long and 15 m broad. Cost of flooring is ₹50 per m². Find the total cost.

Step 1: Find the area of the floor. Area = l × b = 20 × 15 = 300 m²

Step 2: Multiply by cost per m². Total cost = 300 × 50 = ₹15,000

Answer: ₹15,000

Common Errors

• Confusing radius and diameter → Always double-check: diameter = 2r, radius = d/2
• Forgetting π value → Use π = 22/7 unless told otherwise; if answer choices have decimals, use 3.14
• Mixing up CSA and TSA → Curved Surface Area excludes bases; Total Surface Area includes all faces

🔴 Extended

Full Concept

Why πr² is the Area of a Circle Most students just memorize “πr²” without understanding why. Here’s the intuition: imagine a circle as made of countless tiny slices like a pizza, each a thin triangle with height ≈ r and base ≈ arc length. The sum of all bases equals the circumference (2πr). So total area = ½ × r × (sum of all bases) = ½ × r × 2πr = πr². The slices cancel out perfectly. That’s why π is fundamental to circles — it bridges the linear (circumference) and the squared (area).

Heron’s Formula — The Semi-Perimeter Approach For triangles where you don’t know the height, Heron’s formula is a lifesaver. The semi-perimeter s = (a + b + c)/2. Then Area = √[s(s-a)(s-b)(s-c)].

Why does this work? It’s derived from the basic triangle area formula combined with the Pythagorean theorem. The semi-perimeter trick essentially encodes the height calculation into the formula so you never need to find the height explicitly.

Frustum of a Cone A frustum is what you get when you slice the top off a cone. It appears in many practical problems — buckets, lampshades, traffic cones cut off at the top. The slant height l = √[h² + (R - r)²] where R and r are the two radii and h is the vertical height. Surface area of frustum = π(R + r) × l (curved part) + πR² + πr² (both ends).

Cube vs Cuboid — Same Volume, Different Surface Area Here’s a fascinating insight: a cube and a cuboid can have the same volume but very different surface areas. For a given volume V, surface area is minimized when the shape is a cube (all sides equal). This is why cubes are efficient — they pack the most volume with the least surface area. This concept appears in “minimum material to contain X volume” type problems.

Painting Walls Problem When a room’s walls need painting (but not the floor or ceiling), you calculate the wall area as: perimeter × height minus area of doors and windows. Common trap: students forget to subtract the area of openings, or they confuse total wall area with total surface area of all four walls.

Units Conversion Trap This trips up even good students: 1 m = 100 cm. So 1 m² = (100 cm)² = 10,000 cm². Similarly, 1 m³ = 1,000,000 cm³. Always convert to the same unit before doing calculations. If the problem gives area in cm² and asks for cost in ₹/m², convert first.

Combination of Solids Real objects aren’t just one shape — they’re combinations. To find total volume: add volumes of individual parts. For surface area: add curved surface areas, but be careful! When two solids are joined, the surface at the joint disappears from the total surface area. A capsule (cylinder + two hemispheres) has less surface area than the sum of its parts because the flat circular faces where they join are hidden.

Multiple Approaches

Standard: Identify shape → recall formula → substitute values → calculate.

Shortcut: For area of regular polygons you can’t easily split: use the fact that any quadrilateral can be divided into two triangles and use Heron’s formula on each. For related 2D/3D shapes (e.g., a cylinder and a cone on top), use combined volume = V_cylinder + V_cone.

CUET-Level Problems

Q1: A solid metal cylinder of height 10 cm and radius 3 cm is melted and recast into spherical balls of radius 1 cm each. How many balls are formed?

Working: Volume of cylinder = πr²h = π × 9 × 10 = 90π cm³ Volume of one sphere = ⁴⁄₃πr³ = ⁴⁄₃π × 1 = ⁴⁄₃π cm³ Number of balls = 90π ÷ (⁴⁄₃π) = 90 × ³⁄₄ = 67.5 Answer: 67 or 68 (since you need complete balls, 67 full balls can be made, with some metal left over — CUET usually expects you to take integer part: 67)

Q2: The radius of a sphere is increased by 10%. By what percentage does its surface area increase?

Working: Original SA = 4πr² New radius = 1.1r New SA = 4π(1.1r)² = 4π × 1.21 × r² = 1.21 × 4πr² Increase = 21% Answer: 21%

Tricky Cases

• When given slant height but need vertical height for cone volume: Always use l² = h² + r² to find h first. Volume uses h, not l.
• Hemisphere vs half-sphere: A hemisphere is exactly half a sphere — volume = ½ × (⁴⁄₃πr³) = ⅔πr³. But curved surface area = 2πr² (half of 4πr²), while total surface area includes the base circle → 3πr².
• When solid is hollow: Subtract inner volume from outer volume. Example: a pipe is a cylinder with another cylinder removed from inside.
• Units mismatch in area: If sides are in cm and answer options in m², convert cm² to m² by dividing by 10,000 (not 100!).

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