Geometry & Mensuration

Geometry & Mensuration

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