Analytic Geometry
Pythagoras' Theorem and the Cut-and-Paste Proof
The theorem behind every distance calculation, and a proof by rearranging four copies of a triangle inside a square.
What this page covers
- State Pythagoras' theorem precisely, including its hypothesis
- Follow the cut-and-paste proof by area
- Apply the theorem to find an unknown side
- Use the converse to test whether an angle is right
The statement
In any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
Two conditions carry the whole content. The triangle must have a right angle, and c must be the side opposite it. Applying the formula with the wrong side as c gives a wrong answer with no warning.
Always the longest, and always opposite the right angle. If the answer comes out with the hypotenuse shorter than a leg, the sides have been assigned wrongly.
The cut-and-paste proof
The source gives a proof by area that needs no algebra beyond expanding a square. Four copies of the same right triangle are arranged inside a square of side a + b, in two different ways.
- Build the big square. Its side is a + b, so its area is (a + b)2 = a2 + 2ab + b2.
- Arrangement one. Place the four triangles in the corners so that the leftover space is one tilted square of side c. The four triangles have total area 4 × 12ab = 2ab, so the tilted square has area (a + b)2 - 2ab = a2 + b2.
- Arrangement two. Rearrange the same four triangles so the leftover space is two squares, of sides a and b. Their combined area is a2 + b2.
- Conclude. The leftover area is the same in both arrangements, so c2 = a2 + b2.
The 2ab terms cancel, which is the whole trick. The four triangles contribute exactly the cross term of the expansion (a + b)2 = a2 + 2ab + b2, leaving the two squares behind.
The source also draws the classical picture of literal squares erected on the three sides. That is the statement; the rearrangement is the proof.
Using the theorem
Finding a hypotenuse
Legs of 3 and 4.
Finding a leg
Hypotenuse 13, one leg 5.
This is where the hypotenuse condition bites. Adding 52 + 132 gives 194 and a side of about 13.9, longer than the hypotenuse — which is impossible and is the check that catches the error.
| Triple | Check | Multiples |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 6-8-10, 9-12-15, 30-40-50 |
| 5, 12, 13 | 25 + 144 = 169 | 10-24-26 |
| 8, 15, 17 | 64 + 225 = 289 | 16-30-34 |
| 7, 24, 25 | 49 + 576 = 625 | 14-48-50 |
The converse
If the three sides of a triangle satisfy a2 + b2 = c2, then the angle opposite c is a right angle.
This turns the theorem into a test. Measure three lengths, do one arithmetic check, and you know whether a corner is square — which is exactly how a 3-4-5 measurement is used on site.
| Condition | Angle opposite c |
|---|---|
| a2 + b2 = c2 | Exactly 90° |
| a2 + b2 > c2 | Less than 90° (acute) |
| a2 + b2 < c2 | More than 90° (obtuse) |
The last two follow from the cosine rule, c2 = a2 + b2 - 2abcos C: the sign of a2 + b2 - c2 is the sign of cos C, which decides whether C is acute or obtuse.
Where it reappears
Distance formula
D2 = (x2 - x1)2 + (y2 - y1)2 is Pythagoras on the right triangle formed by the coordinate differences.
Equation of a circle
x2 + y2 = r2 says every point on the circle is at distance r from the centre. The source derives it this way.
Trigonometric identity
sin2t + cos2t = 1 is Pythagoras applied to the right triangle inside the unit circle.
Vector magnitude
|v| = √v12 + v22 is the same formula again, applied to components.
Recognising these as one result rather than four is worth the effort: it means an error in any of them can be checked against a triangle drawn on paper.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Adding when a leg is unknown | Subtract | The hypotenuse must be the longest side |
| Using it on a non-right triangle | Use the cosine rule | Confirm the right angle first |
| Taking the wrong side as c | c is opposite the right angle | Mark the right angle on a sketch |
| √a2 + b2 = a + b | No such rule | √9 + 16 = 5, not 7 |
| Forgetting to take the square root | c2 is not c | Check the magnitude against the legs |
Frequently asked questions
Does the theorem work for any triangle?
No. The right angle is essential. For a general triangle the corresponding statement is the cosine rule, c2 = a2 + b2 - 2abcos C, which reduces to Pythagoras when C = 90° because cos 90° = 0.
Why does the cut-and-paste argument prove anything?
Because area is conserved under rearrangement. The same four triangles occupy the same big square in two different arrangements, so whatever space they leave over must be equal in both.
Is the converse also true?
Yes, and it is useful. If a2 + b2 = c2 then the angle opposite c is right. This is how a builder squares a corner with a 3-4-5 measurement.
Where does this show up later?
Everywhere: the distance formula, the equation of a circle, the magnitude of a vector, and the identity sin2t + cos2t = 1 are all Pythagoras in different clothes.
Source. Handwritten teaching notes, Week 4, page 6.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.