Why it is true: the squares really are areas.
a² + b² = c², where c is the hypotenuse.
The theorem is a statement about areas, not lengths. That is why every term is squared, and why you must square before adding.
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- The Pythagorean Theorem
The Pythagorean Theorem
The hypotenuse is always the longest side, and always opposite the right angle.
Find the missing side
A right triangle has a hypotenuse of 13 cm and one leg of 5 cm. How long is the other leg?
The 13 is the hypotenuse, so it goes on its own: 5² + b² = 13², giving 25 + b² = 169 and b² = 144, so b = 12 cm. D adds the squares (25 + 169) as though 13 were a leg — but a leg can never be longer than the hypotenuse, and √194 ≈ 13.9 is. A subtracts the sides instead of their squares.
Finding the hypotenuse: add.
legs 20 and 21
find c
c² = 400 + 441 = 841
square, then add
c = 29
take the root
Square first, add second, root last
20 + 21 = 41 is not the answer, and neither is √20 + √21. The order of operations is what makes the theorem work.
Finding a leg: subtract.
The hypotenuse sits alone on one side of the equation.
hypotenuse 13, leg 5
the opening problem
5² + b² = 13²
13 goes on its own
b² = 169 − 25 = 144
subtract
b = 12
and 12 < 13 ✓
Identify the hypotenuse before writing anything. It is opposite the right angle, and it is always the longest side — so a leg you calculate must come out smaller than it.
Triples worth recognising.
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Any multiple works too: 6-8-10, 9-12-15, 10-24-26. Spotting one saves the whole calculation.
Is 15, 20, 25 a right triangle?
test the converse
225 + 400 = 625
and 25² = 625 ✓
yes — it is 3-4-5 times 5
right-angled
The converse is a genuine test: if a² + b² equals c² the triangle is right-angled. Square the longest side and compare with the sum of the other two squares.
Where it turns up in real problems.
A ladder reaches 2.4 m up a wall
its foot is 0.7 m from the wall
the ladder is the hypotenuse
the wall and ground are the legs
c² = 5.76 + 0.49 = 6.25
c = 2.5 m
Ladders and walls
the ladder is the hypotenuse
it is the slanted one
A rectangle's diagonal
the diagonal is the hypotenuse
sides are the legs
Walking 5 km east, 12 km north
straight-line distance 13 km
shorter than the 17 km walked
When the answer is not a whole number.
Most right triangles are not triples. Leave the answer as a root, or round only at the very end.
A rectangle is 4 cm by 6 cm. How long is its diagonal?
c² = 4² + 6² = 16 + 36 = 52
the sides are the legs
c = √52 ≈ 7.21 cm
52 is not a perfect square
Sense check: 7.21 is longer than 6 but shorter than 4 + 6 = 10 ✓. A calculator gives the decimal; "√52 cm" is the exact answer.
A right triangle has hypotenuse 9 m and one leg 5 m. Find the other leg.
b² = 81 − 25 = 56
finding a leg: subtract
b = √56 ≈ 7.48 m
and 7.48 < 9 ✓
Rounding too early
Rounding √52 to 7 and then using it in a later step drifts the answer. Keep √52 (or all the decimals your calculator shows) until the last line, then round once.
Variations: two equal legs, a height inside a triangle, a triangle that is not right.
The theorem again, where the right triangle has to be found or tested.
An isosceles right triangle has hypotenuse 10 cm. How long is each leg?
a² + a² = 10²
the legs are equal
2a² = 100, a² = 50
a = √50 ≈ 7.07 cm
With equal legs there is one unknown, used twice. Check: 50 + 50 = 100 ✓.
Find the height of an equilateral triangle with sides 6 cm.
the height splits the base: 3 and 3
a right triangle with hypotenuse 6
h² = 6² − 3² = 36 − 9 = 27
finding a leg
h = √27 ≈ 5.20 cm
The right angle is not drawn for us — dropping the height creates it.
Is a triangle with sides 7, 8 and 12 right, acute or obtuse?
7² + 8² = 49 + 64 = 113
the two shorter sides
12² = 144
the longest side
113 < 144
not equal, so not right
When the longest side squared is bigger than the other two squares added, the triangle is obtuse; when it is smaller, acute. So 7, 8, 12 is obtuse.
On the exam
Exams give two sides and ask for the third, ask whether three lengths form a right triangle, or hide the right triangle in a ladder, a rectangle's diagonal or a walk with a turn. The fastest route is to name the hypotenuse first and to know 3-4-5, 5-12-13, 8-15-17 and 7-24-25 and their multiples. For a 10-second check, the hypotenuse must be the longest side and shorter than the two legs added together.
Now you try
Practice problem
Find the diagonal
A rectangular gate is 15 dm wide and 8 dm tall. How long is the diagonal brace across it?
The two sides are the legs and the brace is the hypotenuse: 15² + 8² = 225 + 64 = 289, and √289 = 17 dm. This is the 8-15-17 triple. B subtracts the squares instead of adding. D adds the sides, which is the distance around two edges, not across.
Hint: the diagonal must be longer than either side but shorter than both added together. That alone rules out two options.