Dividing Rectangles & Parallelograms into Minimum Number of Shapes
June 18, 2024 2025-06-18 16:41Dividing Rectangles & Parallelograms into Minimum Number of Shapes
Dividing Rectangles & Parallelograms into Minimum Number of Shapes
Amiya Kumar Dividing Rectangles & Parallelograms into Minimum Number of Shapes
1. Rectangle of Size \( m \times n \) into Minimum Number of Identical Squares
To divide a rectangle of dimensions \( m \times n \) into the minimum number of identical squares:
- Find HCF of \( m \) and \( n \): let it be \( H = \text{HCF}(m, n) \)
- The side of each square is \( H \)
- The total number of squares is:
\[
\text{Number of squares} = \frac{m \times n}{H^2}
\]
2. Rectangle of Size \( m \times n \) into Non-Identical Squares (By Euclidean Algorithm)
This method gives the minimum number of non-identical squares using the process of repeated subtraction/division:
- Each step gives a square of size equal to the remainder
- The total number of squares is the sum of all quotients obtained during division
Use the Euclidean algorithm: count how many times the smaller side goes into the larger, then repeat on the remainder.
3. Parallelogram of Size \( m \times n \) into Minimum Number of Identical Rhombuses
Very similar to rectangle-to-square logic. If the sides are \( m \) and \( n \):
- Find \( H = \text{HCF}(m, n) \)
- Each rhombus will be of side \( H \)
- Number of rhombuses = \( \frac{m \times n}{H^2} \)
4. Parallelogram with Angle = 60° Divided into Equilateral Triangles
When a parallelogram has 60° internal angles, each rhombus can be split into 2 equilateral triangles.
- Step 1: Use the HCF logic to find number of rhombuses
- Step 2: Multiply by 2 to get number of equilateral triangles
\[
\text{Number of equilateral triangles} = 2 \times \left( \frac{m \times n}{H^2} \right)
\]
Use this formula only when the angle of the parallelogram is 60°.
Quick Summary:
- Identical shape division → Use HCF
- Non-identical shape division → Use quotient sum from Euclidean Division
- Equilateral triangles = 2 × rhombuses (only if angle = 60°)
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