Rectangles, Parallelograms & Rhombuses in Grids and Line Systems

Table of Contents

Geometry from Grids and Lines: Rectangles, Parallelograms, and Rhombuses

1. Number of Rectangles Formed by Sets of Parallel Lines (Perpendicular)

If we have two sets of parallel lines—

\(m\) horizontal lines
\(n\) vertical lines

Then each pair of horizontal lines and each pair of vertical lines forms one unique rectangle.

\[ \text{Number of Rectangles} = \binom{m}{2} \times \binom{n}{2} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} \]

2. Number of Parallelograms Formed by Two Sets of Parallel Lines (Non-perpendicular)

If we have:

\(m\) lines in one direction
\(n\) lines in another direction (not perpendicular)

Then, the number of parallelograms formed is the same as for rectangles:

\[ \text{Number of Parallelograms} = \binom{m}{2} \times \binom{n}{2} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} \]

3. Number of Rhombuses Formed (in a Rhombic Grid)

In special cases where both sets of lines are inclined and form equal angles with the horizontal, the parallelograms can become rhombuses.

Again, the formula remains the same:

\[ \text{Number of Rhombuses} = \binom{m}{2} \times \binom{n}{2} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} \]

4. Special Case – Rectangles & Squares in a Grid

In a \(m \times n\) rectangular grid:

🔹 Number of Rectangles

\[ \sum_{i=1}^{m} \sum_{j=1}^{n} i \cdot j = \frac{m(m+1)}{2} \cdot \frac{n(n+1)}{2} \]

🔹 Number of Squares

Only square grids of size \(1 \times 1\), \(2 \times 2\), ..., \(\min(m,n) \times \min(m,n)\) are considered.

\[ \text{Total Squares} = \sum_{k=1}^{\min(m,n)} (m-k+1)(n-k+1) \]

📌 Tip for CAT/XAT Exams: Memorize these combinatorial formulas for quick application. Look for questions involving square grids or inclined lines forming geometric patterns.

To Check More Geometry Questions Click Here

🙏 If these questions helped you, please leave a short review → Click Here

🔔 Subscribe to 🎥 AzuCATion for all video solutions. Click Here

Knowledge Bank

Geometry from Grids and Lines: Rectangles, Parallelograms, and Rhombuses

Geometry from Grids and Lines: Rectangles, Parallelograms, and Rhombuses

Geometry from Grids and Lines: Rectangles, Parallelograms, and Rhombuses

1. Number of Rectangles Formed by Sets of Parallel Lines (Perpendicular)

2. Number of Parallelograms Formed by Two Sets of Parallel Lines (Non-perpendicular)

3. Number of Rhombuses Formed (in a Rhombic Grid)

4. Special Case – Rectangles & Squares in a Grid

🔹 Number of Rectangles

🔹 Number of Squares

How Lines Divide a Plane – A Fundamental Geometry Concept

Dividing Rectangles & Parallelograms into Minimum Number of Shapes

CAT 2025 Offline + Hybrid

Azucation Prep Lab 2025 – Profile Mapping

Address

Explore