Geometric Verification of Pythagoras Theorem
What you will learn
Verify the Pythagorean Theorem (a² + b² = c²) for a right-angled triangle by cutting and rearranging unit squares and algebraic dissection puzzles on graph board.
Materials needed
- Cardboard base sheet (30 cm cm)What is it?Cardboard base sheet (30 cm cm)₹50
Modular acrylic Pythagoras theorem dissection puzzle board with interlocking colored tiles.
Specifications- Base Board
- 300 mm × 300 mm engraved acrylic board
- Tile Materials
- Laser-cut 3 mm colored cast acrylic
- Triples Included
- 3-4-5 and 5-12-13 tile sets
- Precision
- ± 0.05 mm CNC laser cut
- Colored construction paper (red, blue, yellow) sheetsWhat is it?Colored construction paper (red, blue, yellow) sheets₹95
Set of precision drafting tools: stainless steel ruler, right-angle set squares, circular protractor.
Specifications- Ruler
- 30 cm stainless steel with metric graduations
- Set Squares
- 45° and 60° optical acrylic set squares
- Geometry Box (Ruler, Right-angle Set Squares, Pencil)What is it?Geometry Box (Ruler, Right-angle Set Squares, Pencil)₹140
Graph paper sheets and contrasting color craft foam mounting sheets.
Specifications- Grid
- 1 mm precision printed millimeter grid
- Foam Sheets
- 2 mm high-density EVA craft foam
- Pair of Scissors, Glue Stick, and Acrylic Board setWhat is it?Pair of Scissors, Glue Stick, and Acrylic Board set₹185
Standard laboratory apparatus used in Geometric Verification of Pythagoras Theorem.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
- Square grid graph sheet (1 cm cm)What is it?Square grid graph sheet (1 cm cm)₹230
Standard laboratory apparatus used in Geometric Verification of Pythagoras Theorem.
Specifications- Category
- Mathematics Apparatus
- Grade
- CBSE / NEP 2020 Laboratory Standard
Tick items as you collect them. Tap + to see the product, specifications and buy options.
Stay safe
- Ensure all triangle angles are drawn with 90° precision using set squares.
- Cut unit squares strictly along grid lines to ensure edge-to-edge alignment.
- Measure hypotenuse length with high accuracy before cutting the target square.
- Avoid adhesive warping by smoothing paper firmly onto cardboard.
Steps
- 1Draw a right-angled triangle with sides a = 6 cm, b = 8 cm, and hypotenuse c = 10 cm on cardboard.Observation
Cut 4 congruent right-angled triangles with legs a = 3 cm, b = 4 cm, hypotenuse c = 5 cm from colored acrylic card.
SafetyUse steel safety ruler and cutting mat when trimming acrylic card.
- 2Construct square A of dimensions 6 cm × 6 cm (area = 36 cm²) on the base using blue paper.Observation
Construct square of side (a + b) = 7 cm; total area equals (a+b)² = 49 cm².
SafetyVerify all corners are precise 90° angles using set square.
- 3Construct square B of dimensions 8 cm × 8 cm (area = 64 cm²) on the perpendicular using red paper.Observation
Arrange the 4 triangles in corners of the 7×7 cm square; observe central untilted square formed of side c = 5 cm.
SafetyEnsure triangle vertices touch without overlapping.
- 4Construct square C of dimensions 10 cm × 10 cm (area = 100 cm²) on the hypotenuse using yellow paper.Observation
Central square area equals c² = 25 cm²; remaining area equals 4 × (1/2·a·b) = 24 cm²; total = 49 cm².
SafetyRecord measurements in millimeter grid notebook.
- 5Divide square A into 36 unit squares of 1 cm² and square B into 64 unit squares of 1 cm².Observation
Rearrange the same 4 triangles into two rectangles of dimensions a × b; empty spaces form two squares of a² (9 cm²) and b² (16 cm²).
SafetyHandle paper cutouts gently to prevent folding edges.
- 6Cut and rearrange the 36 blue unit squares and 64 red unit squares onto the yellow square C.Observation
Since total area is identical, c² = a² + b² is visually and mathematically demonstrated.
SafetyGlue components onto presentation chart board for permanent record.
- 7Observe that the 100 unit squares from A and B completely and precisely cover square C without any gap or overlap: 36 + 64 = 100.
Tap + on a step to see what you will observe and any safety tips.
Record your observations
| Right Triangle Trial | Leg a (cm) | Leg b (cm) | Calculated c = √(a²+b²) (cm) | Measured c (cm) | Area a² + b² (cm²) | Area c² (cm²) |
|---|---|---|---|---|---|---|
| Triangle 1 (3-4-5 Primitive) | 3.0 cm | 4.0 cm | 5.00 cm | 5.00 cm | 9 + 16 = 25.0 cm² | 25.0 cm² (Exact match) |
| Triangle 2 (5-12-13 Primitive) | 5.0 cm | 12.0 cm | 13.00 cm | 13.00 cm | 25 + 144 = 169.0 cm² | 169.0 cm² (Exact match) |
| Triangle 3 (6-8-10 Scaled) | 6.0 cm | 8.0 cm | 10.00 cm | 10.00 cm | 36 + 64 = 100.0 cm² | 100.0 cm² (Exact match) |
| Triangle 4 (8-15-17 Primitive) | 8.0 cm | 15.0 cm | 17.00 cm | 17.00 cm | 64 + 225 = 289.0 cm² | 289.0 cm² (Exact match) |
| Triangle 5 (7-24-25 Primitive) | 7.0 cm | 24.0 cm | 25.00 cm | 25.00 cm | 49 + 576 = 625.0 cm² | 625.0 cm² (Exact match) |
Tap a highlighted cell to write your own observation.
Result and conclusion
Area of the square on the hypotenuse (c²) exactly equals the combined sum of the areas of squares constructed on the base (a²) and perpendicular (b²) through geometric congruence.
| Observation | What it means |
|---|---|
| Rigid Geometric Conservation | Cutting four identical right triangles (legs a, b) and rearranging them inside an (a+b)×(a+b) square proves c² = a² + b² by Bhaskara dissection. |
| Zero Area Discrepancy | Physical rearrangement of acrylic unit square tiles shows 100% area congruence with zero gaps or overlaps. |
| Pythagorean Triples Verification | Integer triples (3,4,5), (5,12,13), (6,8,10), (8,15,17), (7,24,25) all fulfill the fundamental quadratic metric. |
| Universal Euclidean Validity | Applies universally to any right-angled triangle on a flat Euclidean two-dimensional plane. |
Sample readings. Replace with your own results.