The Pythagorean Theorem · Grade 8
What does Pythagoras' theorem say and why is it true?
The theorem says that the sum of the squares of the two legs equals the square of the hypotenuse. Put another way, the area of the square built on the hypotenuse equals the combined area of the squares built on the legs. It holds for every right triangle without exception, and you can see why using areas.
Learning objectives
- State the theorem in words: the sum of the squares of the legs equals the square of the hypotenuse
- Follow an area based proof of the theorem and explain its key step
- Calculate the hypotenuse when both legs are known
- Recognise the familiar triples three four five and five twelve thirteen
The theorem in words and in areas
In a right triangle, one leg squared plus the other leg squared equals the hypotenuse squared. That is the formula, and it sounds like a rule to memorise until you see what stands behind it.
Build an actual square on each of the three sides. The area of the large one, on the hypotenuse, equals the combined areas of the two smaller ones. The theorem is a statement about areas, which is also where the word 'squared' in the formula comes from.
Why it holds
Take a square whose side is one leg plus the other leg, and fit four copies of the triangle inside it. They leave a tilted square hole in the middle whose side is the hypotenuse.
Now rearrange those same four triangles inside the same big square, differently. This time two holes are left: a square on one leg and a square on the other.
The total area has not changed and the number of triangles has not changed, so what remains must be equal. That is the central step of the proof, and it is why the theorem is a necessity rather than a coincidence.
Triples worth recognising
3, 4, 5 is the famous one: nine plus sixteen is twenty-five. So are 5, 12, 13 and 8, 15, 17, in exactly the same way.
Every multiple of a triple is also a triple. 6, 8, 10 is the same triangle scaled by two, and spotting it saves the whole calculation.
Worked examples
The legs of a right triangle are 3 and 4. What is the hypotenuse?
- Square each leg: 9 and 16
- Add them: 9 plus 16 is 25
- The hypotenuse squared is 25, so the hypotenuse is the root of 25
Answer: The hypotenuse is 5
The legs are 5 and 12. What is the hypotenuse?
- Square each: 25 and 144
- Add: 169
- Take the root: 13 times 13 is 169
Answer: The hypotenuse is 13 — a triple worth knowing
The legs are 2 and 3. What is the hypotenuse?
- Square each: 4 and 9
- Add: 13
- 13 is not a perfect square, so the exact answer is the root of 13
- Estimate: 13 lies between 9 and 16, so the root lies between 3 and 4
Answer: The root of 13, about 3.61
Common mistakes
- Adding the legs and then squaring
- The order is the other way round: square each leg first, then add. Three plus four, squared, is forty-nine, while the correct answer is twenty-five.
- Forgetting to take the root at the end
- The theorem gives the hypotenuse squared, not the hypotenuse. Stopping there returns twenty-five instead of five, which in a question about length is not a length at all.
- Applying the theorem to a triangle that is not right-angled
- It holds only where there is a right angle. Without one the relationship between the sides is different, so look for the right-angle mark before writing the equation.
What to remember
- Leg squared plus leg squared equals hypotenuse squared.
- The theorem is a statement about areas of squares.
- Square first, add second.
- 3-4-5 and 5-12-13 are worth spotting instantly.