📘 Mathematics Subject Enrichment Activity

Chapter: Area

Theme: Transforming Shapes without Changing Area (Śulba-Sūtras)

🔷 Topic

Area Preservation through Geometric Dissection and Rearrangement

🔷 Aim

To understand that area remains the same when a figure is cut and rearranged into another shape, and to explore ancient Indian (Śulba-Sūtras) and Euclidean methods for transforming geometric figures.

🔷 Materials Required

Geometry box (ruler, compass, protractor)
Colour papers / chart paper
Scissors and glue
Pencil & eraser
Colour pencils

🔷 Learning Outcomes

Students will be able to:

Apply area formulas correctly
Perform geometric dissections
Appreciate the contribution of Śulba-Sūtras
Develop visual–spatial reasoning

🔷 Activity 1

Convert a Trapezium into a Rectangle of Equal Area

Procedure

Draw trapezium ABCD.
Extend non-parallel sides to form congruent triangles.
Cut triangles ∆AHI and ∆DGI, ∆BEJ and ∆CFJ.
Rearrange pieces to form rectangle EFGH.

Observation

Cut triangles are congruent.
Rearranged figure is a rectangle.
Area remains unchanged.

Conclusion

Dissection preserves area even though shape changes.

🔷 Activity 1(b)

Construct a Trapezium of Area 144 cm²

Solution (Example):
Take parallel sides = 18 cm, 6 cm
Height = 12 cm

\text{Area} = \frac{1}{2}(18+6)\times12 = 144 \text{ cm}^2

🔷 Activity 2

Convert an Isosceles Trapezium into a Rectangle

Method:
Cut off two congruent triangular ends and shift them inward to form a rectangle.

🔷 Activity 3

Rectangle → Rhombus (Equal Area)

Idea:
Cut along diagonals and rearrange to form a rhombus with same base × height.

🔷 Activity 4

Rhombus → Rectangle (Śulba-Sūtras Method)

Observation:
Height of rhombus becomes breadth of rectangle → area preserved.

🔷 Activity 5

Rectangle → Isosceles Triangle (Śulba-Sūtras)

Method:
Cut rectangle along diagonal and join halves to form triangle.

🔷 Activity 6

Area Comparison

(a) Equilateral Triangle vs Square (same side length a)

Triangle area = $\frac{\sqrt{3}}{4}a^2$
Square area = $a^2$

✅ Square has greater area

(b) Two equilateral triangles vs square

2\times \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{2}a^2 < a^2

✅ Square still has greater area

🔷 Activity 7

Triangle → Rectangle (Śulba-Sūtras)

Method:
Duplicate triangle, join base-to-base → rectangle.

🔷 Activity 8

Isosceles Triangle → Rectangle (Simpler Method)

Hint Used:
∆ADB and ∆ADC become two halves of a rectangle.

🔷 Activity 9

Rectangle of Twice the Area of a Triangle

Method 1:
Duplicate triangle → rectangle

Method 2:
Extend height or base proportionally

🔷 Activity 10

Quadrilateral of Half the Area of Another Quadrilateral

Method:
Divide quadrilateral by diagonal → take one triangle → rearrange.

🔷 Observations (Common)

Shape may change, area does not
Dissection relies on congruency
Visual reasoning is essential

🔷 Reflections (Student Writes)

I learnt that area depends on base and height, not shape.
Ancient mathematicians used logical geometric methods.
Dissection made geometry easier to understand.

🔷 HOTS (Higher-Order Thinking)

Can every polygon be converted into a rectangle of equal area?
Why are triangles most useful for dissection?
How is this concept used in land measurement?

🌟 Mathematical Heritage Link

Śulba-Sūtras (800–500 BCE) show that Indian mathematicians understood area conservation centuries before modern geometry.

1️⃣ Trapezium → Rectangle (Equal Area)

Labels to show on diagram (for students):

Trapezium ABCD
Height h
Parallel sides AB, CD
Cut triangles: ∆AHI ≅ ∆DGI, ∆BEJ ≅ ∆CFJ
Rearranged rectangle EFGH
Same height h

📝 Key Note for Students:

Pieces are rearranged, not resized → area remains same

2️⃣ Isosceles Trapezium → Rectangle

Highlight:

Equal non-parallel sides
Two congruent triangles cut and shifted

3️⃣ Rectangle → Rhombus (Śulba-Sūtras)

Label:

Same base
Same height
Area = base × height (unchanged)

4️⃣ Triangle → Rectangle (Śulba-Sūtras)

Student Hint on Diagram:

Two identical triangles → rectangle

🧠 PART B: Competency-Based Worksheet (with Answer Key)

🔹 Section A: Understanding

What happens to the area when a shape is dissected and rearranged?
Answer: Area remains unchanged.
Name the ancient Indian texts that discuss area transformation.
Answer: Śulba-Sūtras.

🔹 Section B: Application

A trapezium has parallel sides 12 cm and 8 cm, height 10 cm.
Find its area.
Answer:

$\frac{1}{2} (12 + 8) \times 10 = 100 {cm}^{2}$

If this trapezium is converted into a rectangle, what will be the area of the rectangle?
Answer: 100 cm²

🔹 Section C: Reasoning

Why are triangles commonly used in dissection methods?
Answer:
Because triangles are the simplest polygons and can easily form other shapes.
Explain why a rectangle formed from a triangle has the same area.
Answer:
No part is removed or added; only rearranged.

🔹 Section D: HOTS

Can a circle be converted into a rectangle using exact dissection? Why?
Answer:
No, because curved boundaries cannot be exactly rearranged into straight edges.
If the height of a triangle is doubled, what happens to its area?
Answer:
Area doubles.

🔹 Section E: Assertion–Reason

Assertion (A): Area depends on shape.
Reason (R): Rearrangement changes area.

Correct Answer: ❌ Both A and R are false.

📘 PART C: Teacher Rubric Page (Assessment)

Criteria	Excellent (4)	Good (3)	Satisfactory (2)	Needs Improvement (1)
Concept of Area	Clear & accurate	Mostly clear	Partial	Incorrect
Diagrams	Neat, labelled, colourful	Labelled	Rough	Missing
Mathematical Reasoning	Logical & clear	Mostly logical	Limited	Incorrect
Application of Śulba-Sūtras	Correct & explained	Correct	Partial	Not shown
Reflection	Deep insight	Relevant	Minimal	Missing

Total Marks: ____ / 20

🌟 Teacher Tip (Optional Page Note)

This activity integrates history of mathematics, geometry, and visual reasoning, aligning with NEP 2020 competency-based learning.

Friday, February 6, 2026