សិល្បៈ
សម្រស់ និង សុខុមាលភាព
សិប្បកម្ម
វប្បធម៌ និង ប្រវត្តិសាស្ត្រ
ការកម្សាន្ត
បរិស្ថាន
ម្ហូប និង ភេសជ្ជៈ
វិស្វកម្មបញ្ច្រាស
វិទ្យាសាស្ត្រ
កីឡា
បច្ចេកវិទ្យា
ប្រដាប់ដែលស្លៀក
Proving the Pythagorean Theorem by Cutting Squares — a² + b² = c²
The most famous rule in geometry says that in any right-angled triangle, the square built on the longest side equals the two squares on the shorter sides added together: a² + b² = c². The school of Pythagoras proved it around 530 BC. This blueprint proves it the maker's way — with a knotted cord, three cut squares, and a dissection you can hold in your hands. Seeing the pieces of the two small squares fit exactly into the big one is a proof that needs no algebra, and it doubles as the ancient builder's test for a true square corner.
ចាប់ផ្តើម
2
ការណែនាំ
1
1
State the theorem
State the theorem
In a right-angled triangle, name the two short sides a and b and the longest side (the hypotenuse) c. The theorem says the square on c has exactly the same area as the squares on a and b together: a² + b² = c². You will prove this by area, not algebra.
2
2
Make a right angle with a cord
Make a right angle with a cord
Knot a loop of cord into twelve equal spaces. Peg it out as a triangle with sides of 3, 4 and 5 spaces; the corner between the 3 and 4 sides is a perfect right angle. This 3-4-5 trick is how builders have squared corners for thousands of years.
Tools needed:
Cotton Kitchen String3
3
Lay out a right triangle
Lay out a right triangle
Using that right angle, mark a right triangle with legs 3 and 4 units on a board. Its hypotenuse comes out to exactly 5 units — a whole-number right triangle to make the areas easy to count.
Materials for this step:
Red Alder Board1 pieceTools needed:
Chalk Line4
4
Cut the three squares
Cut the three squares
Cut a square on each side of the triangle: 3×3, 4×4 and 5×5. Rule each into unit squares — 9, 16 and 25 of them. These three squares are the whole proof, made physical.
Tools needed:
Hand Saw
Knife5
5
Count the areas
Count the areas
Count: the small squares hold 9 and 16 unit squares, together 25 — exactly the number in the big square on the hypotenuse. 9 + 16 = 25 is a² + b² = c² in plain counting.
6
6
Prove it by dissection
Prove it by dissection
Now cut the two smaller squares into pieces and lay them inside the largest square. They tile it exactly — no gaps, no overlaps. Because the pieces fit for reasons of shape, not luck, this works for every right triangle, not only 3-4-5.
Tools needed:
Knife7
7
Test other triangles
Test other triangles
Repeat with other right triangles: the square on the hypotenuse always equals the sum of the other two. Change the corner so it is no longer a right angle, and the fit fails — which is exactly why the same rule can TEST whether a corner is truly square.
8
8
See why it matters
See why it matters
This single relationship underlies distance, surveying, navigation and building ever since. Any time you find a straight-line distance from two measurements at right angles, you are using Pythagoras — a 2,500-year-old proof you just held in your hands.
សម្ភារៈ
1- 1 pieceកន្លែងទុក
ប្លង់ពាក់ព័ន្ធ
ប្លង់ទាំងនេះចែករំលែកចំណេះដឹង — បច្ចេកទេស សម្ភារៈ ឬគោលការណ៍
CC0 សាធារណៈ
ប្លង់នេះត្រូវបានចេញផ្សាយក្រោម CC0។ អ្នកមានសិទ្ធិចម្លង កែប្រែ ចែកចាយ និងប្រើប្រាស់ដោយមិនចាំបាច់សុំអនុញ្ញាត។
គាំទ្រអ្នកបង្កើតដោយទិញផលិតផលតាមរយៈប្លង់របស់ពួកគេ ដែលពួកគេទទួលបាន កម្រៃជើងសារអ្នកបង្កើត កំណត់ដោយអ្នកលក់ ឬបង្កើតកំណែថ្មីនៃប្លង់នេះ ហើយបញ្ចូលជាការតភ្ជាប់ក្នុងប្លង់របស់អ្នកដើម្បីចែករំលែកចំណូល។