SANAT
GÜZELLİK & SAĞLIK
ZANAAT
KÜLTÜR & TARİH
EĞLENCE
ÇEVRE
YİYECEK & İÇECEK
TERS MÜHENDİSLİK
BİLİMLER
SPOR
TEKNOLOJİ
GİYİLEBİLİR ÜRÜNLER
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.
Başlangıç
2
Talimatlar
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.
Gerekli aletler:
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.
Bu adım için malzemeler:
Red Alder Board1 adetGerekli aletler:
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.
Gerekli aletler:
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.
Gerekli aletler:
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.
Malzemeler
1- 1 adetYer Tutucu
İlgili Blueprint'ler
Bu blueprint'ler bilgi paylaşır — teknikler, malzemeler veya ilkeler
CC0 Kamu Malı
Bu plan CC0 lisansıyla yayınlanmıştır. İzin almadan kopyalayabilir, değiştirebilir, dağıtabilir ve herhangi bir amaçla kullanabilirsiniz.
Planı üzerinden ürün satın alarak Maker'ı destekleyin, böylece Maker Komisyonu Satıcılar tarafından belirlenen komisyonu kazanırlar veya bu Planın yeni bir versiyonunu oluşturun ve gelir paylaşımı için kendi Planınıza bağlantı olarak ekleyin.