Pythagorean Theorem Simulation

About this experiment

Draw a square on each side of a right triangle and the two smaller squares always add up to the largest one. Change the legs, count the unit squares and see the relation hold for every size. Then open or close the angle at C to discover that the equality only works at exactly 90°, which is the converse of the theorem. The rearrangement view shows one of the oldest proofs: the same four triangles fill a big square in two ways, leaving c² in one picture and a² + b² in the other.

Learning goals

State the Pythagorean theorem and identify the hypotenuse of a right triangle
Show with areas that a² + b² = c² for any right triangle
Use the converse to decide whether a triangle is right, acute or obtuse
Follow a visual proof of the theorem by rearrangement

How to use it

1Choose the 3, 4, 5 triangle and count the unit squares in each of the three squares.
2Change a and b with the sliders and check that a² + b² always equals c² while the angle is 90°.
3Move the angle C slider below and above 90°. Compare a² + b² with c² each time.
4Switch to the rearrangement proof and press the button. Explain why the uncovered area does not change.

The model

a^{2} + b^{2} = c^{2}

Pythagorean theorem, where c is the side opposite the right angle

c = a^{2} + b^{2}

Length of the hypotenuse

c^{2} = a^{2} + b^{2} - 2 ab cos C

Law of cosines: the general rule for any angle C

(a + b)^{2} = 4 \cdot \frac{1}{2} ab + c^{2}

Area of the big square in the rearrangement proof

Assumptions

All shapes are drawn exactly on a flat plane. Side lengths can be set in steps of 0.5 units.
Squares are built outward on each side. When a side is not a whole number, the unit grid shows the leftover strip.

Try this in class

Q1A triangle has sides 6, 8 and 10. Is it a right triangle? Check it in the simulation.
Q2If both legs are doubled, what happens to the hypotenuse and to the area of each square?
Q3With a = 3 and b = 4, what angle C gives c = 6? Is the triangle acute or obtuse?
Q4In the rearrangement proof, why must the inner shape in the first picture be a square and not just a rhombus?

Frequently asked questions

Who discovered the theorem?

It is named after Pythagoras, but the relation was known much earlier. Babylonian clay tablets list Pythagorean triples about 1,000 years before him, and the ancient Chinese text Zhoubi Suanjing explains it with a diagram of the 3, 4, 5 triangle.

What is a Pythagorean triple?

Three whole numbers that satisfy a² + b² = c², such as 3, 4, 5 or 5, 12, 13. Any multiple of a triple, like 6, 8, 10, is also a triple.

Does it work for triangles without a right angle?

No. For an acute angle a² + b² is bigger than c², and for an obtuse angle it is smaller. The law of cosines adds a correction term that is zero only at 90°.