The Pythagorean Theorem: Drag a Triangle and Watch the Proof

How to use this simulation

1. Drag the two red points: the right-hand one changes leg a, the top one changes leg b. The sliders do the same.
2. Press ▶ and the coloured “water” in a² and b² drains away while c² fills by exactly the same area. It ends precisely full: nothing spills, nothing is missing.
3. Choose “Rearrangement proof”: four identical triangles slide around inside a big square, and the empty space is c² at one moment and a² + b² at the next. The empty space never changed, so the two are equal.
4. Choose “Converse” and move angle C away from 90°: a² + b² − c² stops being zero, and the poured water either overflows c² (acute) or leaves it short (obtuse).
5. Turn on “Snap to Pythagorean triples” and the triangle only settles on whole-number sides such as 3-4-5, 6-8-10 and 5-12-13; the readings panel then says “Whole-number triple? Yes”.

Ladders, TVs and shortcuts across the field

Picture a ladder leaning on a wall. The builder wants to know how high up the wall it reaches. Nobody climbs up with a tape measure: they measure how far the foot of the ladder is from the wall, they know how long the ladder is, and they work the rest out. That calculation is the Pythagorean theorem.

When a shop sells a “40-inch TV”, 40 inches is neither the width nor the height: it is the diagonal, corner to corner. Knowing the width and height, the same formula gives the diagonal. And when you cut diagonally across the school field instead of walking along two edges, the theorem tells you exactly how many metres you saved.

The theorem carries the name of Pythagoras, a Greek mathematician from about 2,500 years ago, but Babylonian, Egyptian, Indian and Chinese mathematicians used the relationship even earlier. Egyptian surveyors tied 12 evenly spaced knots in a rope and pulled it into a 3-4-5 triangle to get a perfect right angle. So this is not just an exam formula; it has been a working tool for thousands of years.

What is the Pythagorean theorem? Statement and formula

The textbook statement: “In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.” When an exam says “state the Pythagorean theorem”, write exactly that sentence.

In plain words: if one angle of a triangle is exactly 90°, the side opposite that angle (always the longest side) is the hypotenuse. The two sides that form the right angle are called the legs, or the base and the perpendicular. Draw a square on each of the three sides. The areas of the two small squares add up to exactly the area of the big one.

Take the simulation's starting triangle: a = 3, b = 4. Then a² = 9 and b² = 16, which add up to 25. The hypotenuse is c = 5, so c² = 25. Turn on the unit-square tiles and count: 9 squares in the green one, 16 in the orange one, and exactly 25 in the big one.

Proof 1: four triangles in a square

This is the proof moving in the simulation's “Rearrangement proof” mode. In an exam, set it out like any theorem: the given, the construction, the proof, with a reason beside every step.

Proof 2: the trapezium (Garfield's proof)

Take right triangle ABC with the right angle at B, BC = a, AB = b and AC = c. Extend BC to D so that CD = b, and draw DE perpendicular to BD with DE = a. Join A to E.

ABDE is a trapezium, because AB and DE are both perpendicular to BD and therefore parallel. Its parallel sides are b and a, and its height is BD = a + b. Inside it sit three triangles: ABC, CDE and ACE. The first two are congruent, and angle ACE is 90° because the two angles beside it at C add to 90°.

Now compare areas. The trapezium has area ½(a + b)(a + b). The three triangles have ½ab + ½ab + ½c². Setting them equal and doubling gives (a + b)² = 2ab + c², so a² + b² = c². Fun fact: James Garfield, later the 20th President of the United States, published this proof in 1876.

Proof 3: similar triangles

In right triangle ABC with ∠C = 90°, drop the perpendicular CD from C onto the hypotenuse AB. It splits the triangle into two smaller ones, and all three triangles are similar: each has a right angle and shares one angle with the big triangle.

Similarity gives AC² = AD · AB and BC² = BD · AB. Adding, AC² + BC² = AB(AD + BD) = AB · AB = AB². This is the proof most higher-level textbooks use, including India's NCERT books, and it is the one that generalises to the rest of geometry.

The converse: does it work backwards?

The converse of the Pythagorean theorem says: if the square on one side of a triangle equals the sum of the squares on the other two sides, then the angle between those two sides is a right angle. So you can decide whether a triangle is right-angled from its three side lengths alone.

Even when the sides do not balance, you learn something. If a² + b² > c², the angle opposite c is less than 90° and the triangle is acute. If a² + b² < c², that angle is more than 90° and the triangle is obtuse. The reason is simple: open the angle and the opposite side gets longer; close it and the side gets shorter.

The “Converse” mode shows exactly this. Set C to 120°: c grows, and even all the water from a² and b² cannot fill c². Set C to 60°: c shrinks and the water overflows, turning red. Only at exactly 90° does it fill c² to the brim.

Pythagorean triples: when all three sides are whole numbers

Most right triangles have a messy hypotenuse: legs of 1 and 1 give √2 ≈ 1.414. A few special ones have three whole-number sides, and such a set is called a Pythagorean triple. The famous one is 3, 4, 5.

Multiply a triple by any whole number and you get another: 3-4-5 gives 6-8-10, 9-12-15 and 12-16-20. There is also a formula that makes new ones (Euclid's formula): pick whole numbers m > n, and m² − n², 2mn and m² + n² always form a triple. Every row of the table below is computed, and the last column checks it.

The distance formula is Pythagoras in disguise

You do not need to memorise the distance formula between (x₁, y₁) and (x₂, y₂); you can build it. The horizontal gap x₂ − x₁ and the vertical gap y₂ − y₁ are the legs of a right triangle, and the straight-line distance between the points is its hypotenuse.

In the same way, the length of a vector (|v| = √(x² + y²)), the diagonal of a box in three dimensions and even the trigonometric identity sin²θ + cos²θ = 1 are all the Pythagorean theorem wearing different clothes. Understand this chapter well and a large part of coordinate geometry, physics and admission-test maths becomes easy.

Experiments to try in the simulation

Reading a theorem does not make it stick; changing things with your own hands does. Try these one by one and note what you see.

• Halve both a and b: c halves too, but every square's area drops to a quarter. Notice the difference between length and area.
• Keep a fixed and keep increasing b: c grows, but it never reaches a + b. Any two sides of a triangle add up to more than the third.
• Make a = b (an isosceles right triangle): the hypotenuse is a × √2, and the square on c is exactly two of the small squares.
• In rearrangement mode, change a and b: the empty space is equal in both arrangements for every size. One example is not a proof; working for every triangle is.
• In converse mode, sweep angle C slowly past 90° and watch the “a² + b² − c²” reading go from positive, through zero, to negative.

Solved examples

Every number below is calculated in this page's code from the formula. Work each one in your notebook and compare.

Mistakes everyone makes

Marks are rarely lost for not knowing the formula; they are lost to these small slips.

• Calling the wrong side c. The hypotenuse is always opposite the right angle and always the longest; find it first.
• Writing c = a + b. Adding squares is not adding sides: 3 + 4 = 7, but the hypotenuse is 5.
• Forgetting the square root: getting c² = 25 and writing 25 as the answer. The last step is always √.
• Using the formula on a triangle that is not right-angled. The condition is one angle of exactly 90°; otherwise you need the law of cosines.
• Leaving out the reasons in a proof. Steps without the “why” do not earn full marks.
• Mixing units: one side in centimetres, another in metres. Convert first.

The Pythagorean theorem in real life

Architects and builders check square corners with 3-4-5. Surveyors measure diagonal distances across land this way. For short distances, a map app finding how far apart two places are is essentially doing this sum. In video games, the distance between two characters, or between the mouse and a button, is √(dx² + dy²).

In physics, the resultant of two perpendicular velocities or forces comes from Pythagoras: the true speed of a boat crossing a river, the speed of a projectile at any instant. In electrical engineering, the impedance of an AC circuit uses the same formula. It is not just a geometry chapter; it is one of the most used formulas in all of mathematics.

Exam corner

In middle school and Class 8 to 10 exams, expect: find the hypotenuse or a leg; state and prove the theorem; decide whether a triangle is right-angled; and a word problem about a ladder, a pole, a field or a screen. In Bangladesh's SSC and in many Indian state boards the proof itself is a common long question; check your current syllabus, since CBSE's rationalised syllabus changed how the proof is treated.

In higher classes the theorem rarely appears by name, but it is everywhere: the distance formula, vector magnitudes, trigonometric identities and the law of cosines. In admission and entrance tests, recognising a Pythagorean triple instantly saves real time.

• Learn the statement word for word: “In a right-angled triangle, the square on the hypotenuse …”.
• In a proof, draw the figure, write the given and the construction, and give a reason for every step.
• Practise at least two proofs (four triangles and the trapezium); a paper may ask for “another proof”.
• Always write units, and give a non-whole square root to two decimal places.

Revision summary

Read just this part the night before the exam and the whole chapter comes back.

• Condition: one angle of the triangle is exactly 90°.
• Formula: hypotenuse² = leg² + leg², a² + b² = c².
• Hypotenuse: c = √(a² + b²); leg: a = √(c² − b²).
• Converse: a² + b² = c² means right; greater means acute; smaller means obtuse.
• Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25; Euclid: m² − n², 2mn, m² + n².
• Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²].

Frequently asked questions