Rhombus (समचतुर्भुज) - Complete Geometry Guide

A Rhombus is a special type of quadrilateral that plays an important role in Geometry, especially in competitive exams like RRB NTPC, Group D, and SSC.

✨ Definition (परिभाषा)

A rhombus is a four-sided figure (quadrilateral) with all sides equal and diagonals that bisect each other at right angles (90°).

In Hindi:

समचतुर्भुज एक ऐसा चतुर्भुज होता है जिसकी सभी भुजाएँ बराबर होती हैं और विकर्ण एक-दूसरे को समद्विभाजित करते हैं तथा 90° पर काटते हैं।

🔹 Properties of a Rhombus

Property	Explanation
All sides equal	AB = BC = CD = DA
Opposite angles equal	∠A = ∠C, ∠B = ∠D
Diagonals intersect at 90°	AC ⊥ BD
Diagonals bisect each other	Each diagonal cuts the other into two equal halves
Diagonals bisect angles	Each diagonal divides vertex angles equally

📚 Area Formula (क्षेत्रफल का सूत्र)

The area of a rhombus is given by the product of its diagonals divided by 2:

\text{Area} = \frac{1}{2} \times d_1 \times d_2

Where:

( d_1 ) = length of diagonal 1 (AC)
( d_2 ) = length of diagonal 2 (BD)

In Hindi:

\text{समचतुर्भुज का क्षेत्रफल} = \frac{1}{2} \times \text{विकर्ण}_1 \times \text{विकर्ण}_2

📈 Example Question (परीक्षा में आने वाला सवाल)

Q. Diagonals of a rhombus are 10 cm and 8 cm. Find its area.

Solution:

\text{Area} = \frac{1}{2} \times 10 \times 8 = 40\ cm^2

✅ Answer: 40 cm²

📊 Rhombus vs Other Quadrilaterals

Shape	Area Formula	Diagonal Property
Square	side² or 1/2 × d₁ × d₂	Equal diagonals at 90°
Rectangle	l × b	Equal diagonals
Rhombus	1/2 × d₁ × d₂	Unequal diagonals at 90°
Parallelogram	b × h	No specific diagonal rule

🛫 Other Formulas

Perimeter of Rhombus:

4 \times \text{Side}

If diagonals are known and side is needed:

\text{Side} = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}

(This is derived using the Pythagoras Theorem.)

🎓 Important Keywords for Exams

Rhombus = समचतुर्भुज
Diagonal = विकर्ण
Area = क्षेत्रफल
Side = भुजा
Bisection = समद्विभाजन
Right angle = समकोण

📆 Exam Tips (RRB NTPC / Group D)

Area formula is very frequently asked.
Practice finding area from diagonals directly.
Remember that diagonals intersect at 90°, even if they are unequal.
Expect 1-2 questions from mensuration in Stage 1 and 2.

🎨 Visual Diagram (Representation)

         D
        / \ 
     d1/   \d1
      /     \     
     A-------C
     |       |
  d2 |       | d2
     |       |
     B-------B

Note: Diagonals AC and BD intersect at 90° and divide the rhombus into 4 right-angled triangles.

✍️ Practice Makes Perfect!

Try solving 3–5 rhombus-based questions to solidify your understanding.