Curriv

Pythagoras Theorem

Chapter 2 · Mathematics · Class 8 30 min read

Why This Matters

Picture a ladder leaning against a wall. Its foot is 6 metres out from the wall, and its top reaches 8 metres up. How long is the ladder?

You cannot stretch a measuring tape along a ladder that is high in the air. But you can measure along the flat ground, and you can measure up the wall. From just those two numbers, there is a way to know the ladder’s length exactly — without ever touching the ladder.

That “way” is one of the oldest and most useful rules in all of mathematics. It connects the three sides of a right-angled triangle. Once you know it, you can find a hidden distance from two known ones: the length of a ramp, the diagonal of a TV screen, the straight-line distance between two points on a map, the height of a tree from its shadow and the slant of a string.

The rule was written down in India by a scholar named Baudhāyana around 800 BCE — long before the Greek thinker Pythagoras. So it is fairly called the Baudhāyana-Pythagoras theorem. By the end of this chapter you will understand why it is true, not just that it is true — and you will be able to use it with confidence.

The Big Idea

In a right-angled triangle, the square on the longest side is exactly equal to the squares on the other two sides added together. If the two shorter sides are a and b, and the longest side (the hypotenuse) is c, then a² + b² = c². This one short rule lets you find any one side of a right triangle as soon as you know the other two.

Let’s Break It Down

The right triangle and its hypotenuse

A right-angled triangle (or just “right triangle”) is a triangle with one angle equal to exactly 90° — a perfect square corner, like the corner of this page. We mark that corner with a small square.

The two sides that form the right angle are called the legs. The third side — the one opposite the right angle — has a special name: the hypotenuse. It is always the longest side of the triangle.

Before we go further, let us quickly refresh what “square of a number” and “square root” mean, since the whole chapter leans on them.

Figure 2.1 below names the three sides of a right triangle and marks the right angle.

A right triangle with the right angle at the bottom-left corner. The horizontal leg is labelled a, the vertical leg is labelled b, and the long slanting side opposite the right angle is the hypotenuse labelled c.
Figure 2.1 — The parts of a right-angled triangle. The corner at C carries a small red square and the label 90 degrees, showing it is a right angle. The two sides that meet at this corner are the legs, labelled a (the horizontal leg) and b (the vertical leg). The third side, opposite the right angle, runs from A to B and is the hypotenuse, labelled c. The hypotenuse is always the longest side. The theorem in this chapter links these three sides.

So whenever you meet a right triangle, find the right angle first. The side facing it, across the triangle, is the hypotenuse c. The other two are the legs a and b.

Concept check

In a right triangle, which side is the hypotenuse, and how do you spot it?

Baudhāyana’s clever start: doubling a square

Long ago, Baudhāyana asked a simple-sounding question while designing fire altars: how do you make a square with exactly double the area of a given square?

A first guess is to double the length of each side. But that is wrong, and it is worth seeing why. If you double each side, you do not double the area — you make it four times bigger. A 2×2 square has area 4; a 4×4 square has area 16, which is 4 times as much, not 2 times. Doubling the side multiplies the area by 2 × 2 = 4.

So how do you double the area? Baudhāyana’s beautiful answer: build the new square on the diagonal of the old one.

Here is the reason, and it is the key idea behind the whole theorem. Let’s see it in a picture first.

On the left a small square is cut by both diagonals into four equal triangles. On the right a larger tilted square stands on the diagonal of the small one and is made of four of those same triangles, so it has double the area.
Figure 2.2 — Baudhayana's doubling of a square. Panel (a) shows the original square cut by its two diagonals into 4 equal triangles; each half of the square is made of 2 of them, so the whole square is 2 plus 2. Panel (b) shows the square built on the diagonal of the original square. It is split the same way and is made of 4 of the very same triangles. So the diagonal square contains 4 triangles against the original's 2 — it has exactly double the area. The red arrow links the two squares.

Look carefully. The original square is made of 2 of these triangles in each half — but let us count the whole square as built from triangles. The square on the diagonal is made of 4 of the same triangles. Four against two: the diagonal square is exactly double the area. No measuring needed — just counting equal triangles.

This is the seed of everything. It already tells us something about a special right triangle.

The isosceles right triangle: a first taste

An isosceles triangle has two equal sides. An isosceles right triangle has a right angle and its two legs equal. Take a square of side 1. A diagonal cuts it into two isosceles right triangles, each with two legs of length 1.

What is the length of the hypotenuse (the diagonal)? Call it c. The square built on that diagonal has double the area of the unit square, so its area is 2. But the area of a square of side c is c². So:

c² = 2 × (area of unit square)

c² = 2 × 1 = 2

So c = √2 (a number a little more than 1.4)

So the diagonal of a 1×1 square is √2. This number √2 is about 1.414, and its digits go on for ever without repeating — but that is a story for a later class. For now, notice what just happened: we found a side length using only areas of squares. That is the trick we are about to make general.

Concept check

If you double the length of every side of a square, how many times bigger does its area become?

The theorem: a² + b² = c²

Now the big result. It works for any right triangle, not just the equal-legged one.

Baudhāyana-Pythagoras theorem: In a right-angled triangle with legs a and b and hypotenuse c,

a² + b² = c²

In words: the square on the hypotenuse equals the sum of the squares on the two legs. Remember, “the square on a side” means a real square drawn on that side, and its area is the side length times itself.

Let us see it before we prove it. We will use the most famous right triangle of all, with sides 3, 4 and 5.

A 3-4-5 right triangle drawn to scale on a grid, with a square of cells on each side. The square on the 3-leg has 9 cells, the square on the 4-leg has 16 cells, and the square on the 5-hypotenuse has 25 cells.
Figure 2.3 — Squares on the sides of a 3-4-5 right triangle, all drawn to the same scale. The yellow triangle in the middle is right-angled, with legs 3 and 4 and hypotenuse 5. The blue square sits on the 3-leg and is divided into 9 small cells, so its area is 3 squared equals 9. The green square on the 4-leg holds 16 cells, so its area is 4 squared equals 16. The purple square on the hypotenuse holds 25 cells, so its area is 5 squared equals 25. Count them: 9 plus 16 equals 25. The two smaller squares together have exactly as many cells as the big square. That is a squared plus b squared equals c squared.

Count the cells. The square on leg 3 has 9 cells. The square on leg 4 has 16 cells. Together that is 9 + 16 = 25 cells — exactly the number of cells in the square on the hypotenuse 5, which is 5² = 25. The two small squares, added, fill the big square perfectly.

But one picture of one triangle is not a proof. Why should this work for every right triangle? Let us prove it.

Why it is true: the four-triangles proof

This proof is short, visual, and convincing. Here is the plan: take four copies of the same right triangle and fit them inside a big square in two different ways. Figure 2.4 shows the arrangement.

A big square of side a plus b holds four identical right triangles with legs a and b in its corners. The space left in the middle is a tilted square whose side is the hypotenuse c. A side panel shows the area calculation giving c squared equals a squared plus b squared.
Figure 2.4 — A proof that a squared plus b squared equals c squared. The big yellow square has side a plus b. Four identical right triangles (legs a and b, drawn blue and green) are tucked into its four corners. The space they leave in the middle is a tilted purple square whose side is the hypotenuse c, so its area is c squared. The side panel does the area bookkeeping: the big square is (a plus b) squared; the four triangles together have area 2ab; what is left is c squared. So c squared equals (a plus b) squared minus 2ab, which simplifies to a squared plus b squared.

Follow the area count. The big square has side (a + b), so its area is (a + b)².

Inside it sit four copies of our right triangle. Each triangle has area ½ × a × b (half of base times height), so four of them have area 4 × ½ab = 2ab.

The shape left in the middle is a tilted square. Each of its sides is a hypotenuse of length c, so its area is c².

Now the leftover (the tilted square) is the big square minus the four triangles:

c² = (a + b)² − 2ab

c² = a² + 2ab + b² − 2ab (because (a + b)² = a² + 2ab + b²)

c² = a² + b²

The two +2ab and −2ab cancel out, and we are left with a² + b² = c². That holds for any a and b, so the theorem is true for every right triangle. Beautiful, isn’t it?

Finding a missing side

Once you trust a² + b² = c², you can find any one side from the other two. There are two cases.

Case 1 — you have both legs and want the hypotenuse. Add the squares of the legs, then take the square root.

Find the hypotenuse

A right triangle has legs of length 3 cm and 4 cm. Find the hypotenuse c.

Case 2 — you have the hypotenuse and one leg, and want the other leg. Here you must subtract. Rearrange a² + b² = c² into b² = c² − a².

Find a missing leg

A right triangle has one leg of 8 cm and a hypotenuse of 17 cm. Find the other leg.

The big thing to watch: the hypotenuse is always c, the longest side. If a problem gives you the longest side, you are in the subtract case. If it gives you the two shorter sides, you are in the add case.

Concept check

You are given the hypotenuse and one leg of a right triangle. To find the other leg, do you add or subtract the squares?

Pythagorean triples: whole-number triangles

Notice something nice about the 3, 4, 5 triangle: all three sides are whole numbers, with no fractions or square-root signs. Such a set of three whole numbers that fits a² + b² = c² is called a Pythagorean triple (Baudhāyana listed many of these, so they are also called Baudhāyana triples).

The classic ones are (3, 4, 5) and (5, 12, 13). Figure 2.5 draws both to scale and shows how to make many more.

The 3-4-5 triangle and the 5-12-13 triangle both drawn to scale as right triangles with whole-number sides. A panel shows that multiplying a triple by a whole number, like 3-4-5 times 2 giving 6-8-10, makes another triple.
Figure 2.5 — Pythagorean (Baudhayana) triples. Panel (a) is the 3-4-5 right triangle: 3 squared plus 4 squared is 9 plus 16, which is 25, which is 5 squared. Panel (b) is the 5-12-13 right triangle: 5 squared plus 12 squared is 25 plus 144, which is 169, which is 13 squared. The purple panel on the right shows you can make endless new triples by multiplying every number of a triple by the same whole number: 3-4-5 times 2 gives 6-8-10, times 3 gives 9-12-15, and so on for ever.

Why does multiplying work? Take (3, 4, 5) and multiply each number by, say, 2 to get (6, 8, 10). Check it: 6² + 8² = 36 + 64 = 100 = 10². It works! The reason is that squaring a multiplied number just multiplies the square by that factor squared, and the factor cancels on both sides. So from one triple you get infinitely many.

A triple with no common factor bigger than 1 — like (3, 4, 5) or (5, 12, 13) — is called a primitive triple. The others, like (6, 8, 10), are just scaled copies. Knowing a few primitive triples by heart (3-4-5, 5-12-13, 8-15-17, 7-24-25) makes many problems instant.

The converse: a test for a right angle

So far we used the theorem forwards: “this triangle is right-angled, so a² + b² = c².” It also works backwards, and this is incredibly handy:

Converse: If the three sides of a triangle satisfy a² + b² = c² (with c the longest side), then the triangle is right-angled — and the right angle is opposite the longest side.

This gives a quick test for a right angle using only a ruler. Builders use a version of it: a rope knotted into a 3-4-5 triangle makes a perfect square corner with no protractor. Figure 2.6 shows the test passing and failing.

On the left a 6-8-10 triangle where 6 squared plus 8 squared equals 10 squared, so it has a true right angle. On the right a 4-6-8 triangle where 4 squared plus 6 squared is 52 but 8 squared is 64, so it is not right-angled.
Figure 2.6 — The converse used as a test. On the left, the green triangle has sides 6, 8 and 10. Checking: 6 squared plus 8 squared is 36 plus 64, which is 100, exactly 10 squared. The test passes, so the triangle is right-angled (the right angle is marked). On the right, the red triangle has sides 4, 6 and 8. Checking: 4 squared plus 6 squared is 16 plus 36, which is 52, but 8 squared is 64. Since 52 is not 64, the test fails and this triangle has no right angle. Always square the two shorter sides and compare with the square of the longest.

To use the test: square all three sides, add the squares of the two shorter sides, and see if you get the square of the longest side. Equal means right-angled; not equal means not.

Is it a right triangle?

A triangle has sides 9 cm, 12 cm and 15 cm. Is it a right-angled triangle?

A real-life distance: the ladder

Now back to the ladder from the start of the chapter. The wall, the ground, and the ladder make a right triangle. The right angle is where the wall meets the ground. The ladder is the slanting side — the hypotenuse.

A ladder leans against a wall, forming a right triangle. The foot is 6 metres from the wall, the top reaches 8 metres up, the corner where the wall meets the ground is a right angle, and a panel works out the ladder length as 10 metres.
Figure 2.7 — Using the theorem in real life. The wall, the ground and the ladder form a right triangle. The foot of the ladder is 6 metres out along the ground (this is leg a), the top reaches 8 metres up the wall (leg b), and the small red square shows the right angle where the wall meets the ground. The ladder itself is the hypotenuse c. The green panel works it out: c squared is 6 squared plus 8 squared, which is 36 plus 64, equal to 100, so c is the square root of 100, which is 10 metres. The ladder is 10 metres long.

The legs are 6 m and 8 m, so:

c² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 m

The ladder is 10 metres long — found from two ground-level measurements, just as promised.

This same idea finds the straight-line (“crow-flies”) distance between two points when you know how far apart they are east-west and north-south. Walk 6 km east and 8 km north, and you are exactly 10 km in a straight line from where you started. The Pythagoras theorem is the bridge between flat measurements and slant distances everywhere — screens, ramps, maps, and roofs.

Common Mistakes

Read these three slips once and you will avoid the traps that cost most students marks.

⚠️ Common mistake
What students think

In a² + b² = c², you can put any of the three sides as c.

Why it seems right

The letters a, b, c look interchangeable, like in other formulas, so it feels like it should not matter which side you call c.

What actually happens

c must always be the hypotenuse — the longest side, opposite the right angle. The two legs go in a and b. If you put a leg where c belongs, the sum comes out wrong. Always find the right angle first and make the side opposite it your c.

⚠️ Common mistake
What students think

To find a missing leg, you add the squares of the two sides you are given.

Why it seems right

The theorem is famous as 'add the squares', so students add every time without checking which side is missing.

What actually happens

You only add when both given sides are legs and the hypotenuse is missing. If the hypotenuse is one of the given sides, you must subtract: b² = c² − a². Check whether the unknown is the longest side (add) or a leg (subtract) before you start.

⚠️ Common mistake
What students think

After getting c² = 25, the answer is c = 25.

Why it seems right

The number 25 is sitting right there after the addition, so it is tempting to stop and call it the side length.

What actually happens

c² = 25 means c squared is 25, not c itself. You must take the square root to undo the square: c = √25 = 5. The side is 5, not 25. Always finish by square-rooting to get the actual length.

Quick Check

Try each one. They check the key ideas of the chapter.

In a right triangle, which side is the hypotenuse?

A right triangle has legs 5 cm and 12 cm. What is its hypotenuse?

A triangle has sides 8, 15 and 17. Is it right-angled?

If c² = 100, what is c?

Practice Problems

Try each one yourself first. Only then tap to see the full solution.

Easy

easy

A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.

easy

Is (7, 24, 25) a Pythagorean triple? Check using a² + b² = c².

Medium

medium

A right triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg.

medium

Find the length of the diagonal of a square whose side is 5 cm.

medium

A lotus flower stands 1 unit above the surface of a still lake. A breeze pushes its tip until it just touches the water 3 units away from where it started. How deep is the lake at that spot? (This is a classic problem from Bhāskarāchārya's Lilavati.)

Challenge

challenge

A rhombus has diagonals of length 24 units and 70 units. Find the length of each side. (Hint: the diagonals of a rhombus cross at right angles and cut each other in half.)

challenge

Show that the hypotenuse must be the longest side of any right triangle.

Summary

  • A right-angled triangle has one 90° angle. The two sides forming it are the legs (a and b); the side opposite it is the hypotenuse (c), always the longest side.
  • The Baudhāyana-Pythagoras theorem says a² + b² = c²: the square on the hypotenuse equals the sum of the squares on the two legs.
  • It can be proved by fitting four copies of the same right triangle into a square of side (a + b); the leftover middle square has area c², which works out to a² + b².
  • To find a side: if both legs are known, add the squares and square-root for the hypotenuse; if the hypotenuse and one leg are known, subtract the squares for the other leg.
  • A Pythagorean (Baudhāyana) triple is three whole numbers with a² + b² = c² — like (3, 4, 5), (5, 12, 13), (8, 15, 17). Multiplying a triple by any whole number gives another triple, so there are infinitely many.
  • The converse is a test: if a² + b² = c² holds (c the longest side), the triangle is right-angled. Builders use a 3-4-5 rope to make a perfect square corner.
  • The theorem turns flat measurements into slant distances — ladders, diagonals, screens, and straight-line (“crow-flies”) distances on a map.

What’s Next

You can now find any side of a right triangle, test whether an angle is a right angle, and you have seen a real proof of why the theorem is true — not just memorised it. That habit of asking “but why?” is what real mathematics is about.

Next we return to comparing quantities and scaling them up and down. The next chapter is Proportional Reasoning - 2, where you will use ratios and proportions to solve everyday problems about speed, price, and sharing. See you there!

Frequently Asked Questions

What is the Pythagoras theorem in simple words?

In any right-angled triangle, the square on the longest side equals the sum of the squares on the other two sides. If the two shorter sides are a and b, and the longest side (the hypotenuse) is c, then a squared plus b squared equals c squared. It lets you find one side when you know the other two.

Why is it also called the Baudhayana theorem?

An Indian scholar named Baudhayana stated this rule in his Sulba-Sutra around 800 BCE, hundreds of years before the Greek thinker Pythagoras. Baudhayana was the first in history to write it in a clear, general form, so the rule is fairly called the Baudhayana-Pythagoras theorem.

What is the hypotenuse of a right triangle?

The hypotenuse is the side directly opposite the right angle. It is always the longest side of a right-angled triangle. In the rule a squared plus b squared equals c squared, the hypotenuse is c, the side whose square equals the sum of the other two squares.

What is a Pythagorean triple?

A Pythagorean triple is a set of three whole numbers a, b and c that fit a squared plus b squared equals c squared, so they can be the sides of a right triangle with no fractions. The most famous is 3, 4, 5. Others are 5, 12, 13 and 8, 15, 17. Multiplying every number in a triple by the same whole number gives another triple.

How do you check if a triangle is right-angled?

Use the converse of the theorem. Square each of the three sides. Add the squares of the two shorter sides. If that sum equals the square of the longest side, the triangle is right-angled. If it does not, the triangle has no right angle.