Mensuration (2D)

Mensuration (2D)

Concept Explanation

Mensuration is just a fancy word for “measuring shapes.” When you want to know how much space a flat shape covers — like how much carpet you need for a room — you’re looking for its area. When you want to know how far it is to walk around the edge — like how much fencing you need for a plot of land — you’re looking for its perimeter or circumference.

The four shapes that show up most often in exams are squares, rectangles, circles, and triangles. Each one has its own personality. A square is the simplest — all four sides are identical, so you only need to know one number (the side length) to figure out everything about it. A rectangle is like a stretched square — opposite sides match, so you need two numbers: length and breadth. A circle is different because it has no corners at all; everything revolves around one special number called the radius, which is the distance from the center to any point on the edge. A triangle is the simplest polygon — just three sides — and its area depends on how tall it is (the height), measured straight up from the base.

Key Formulas

Symbol	Meaning
a	Side of a square
l	Length of a rectangle
b	Breadth of a rectangle (or base of a triangle)
r	Radius of a circle
d	Diameter of a circle (d = 2r)
h	Height of a triangle
π	Pi, approximately 3.14159 or 22/7

Step-by-Step Example

Q: The radius of a circle is 7 cm. Find its area and circumference.

Step 1: Identify the formula for area of a circle. Area = πr²

Step 2: Substitute r = 7. Area = π × 7² = π × 49 = 22/7 × 49 = 154 cm²

Step 3: Identify the formula for circumference. Circumference = 2πr

Step 4: Substitute r = 7. Circumference = 2 × π × 7 = 2 × 22/7 × 7 = 44 cm

Answer: Area = 154 cm², Circumference = 44 cm

Common Mistakes

• Using diameter instead of radius in the area formula → Always square the radius first, not the diameter. If given diameter, halve it first.
• Confusing area with perimeter → Area is inside (measured in square units), perimeter is the boundary walk (measured in linear units).
• Forgetting to use consistent units → If length is in cm, area will be in cm². Mixing meters and centimeters destroys accuracy.

Quick Test (2 Qs)

1. Q: A rectangle’s length is 12 cm and its diagonal is 13 cm. What is its area? Options: A) 30 cm² B) 60 cm² C) 78 cm² D) 156 cm². Ans: B (Reason: breadth² = 13² – 12² = 169 – 144 = 25, so breadth = 5; area = 12 × 5 = 60 cm²)

2. Q: The circumference of a circle is 44 cm. What is its area? Options: A) 77 cm² B) 154 cm² C) 308 cm² D) 616 cm². Ans: B (Reason: 2πr = 44 → r = 7 cm; area = πr² = 22/7 × 49 = 154 cm²)