Pythagorean Theorem: Discover the Missing Side
Understand that in a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides, and that the rule can be rearranged to find any missing side.
Before you try
What if the triangle had no right angle? Would the squares still add up?
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This rule works only for right-angled triangles. Triangles without a right angle need different rules.
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Try these ideas
- Set a = 3 and b = 4 and check that 9 plus 16 makes 25.
- Turn on the unit grid and count the squares inside each shape.
- Switch to Find the missing side and solve five problems.
Why it works
Draw a square on each side of a right-angled triangle and something remarkable happens: the areas of the two smaller squares add up to exactly the area of the largest one. That is the Pythagorean theorem, written a² + b² = c², where c is the hypotenuse, the longest side and the one opposite the right angle. It only works when there is a right angle, which is why the little square marker in the corner matters.
To find the hypotenuse you add the squares and take the square root. To find a missing short side you subtract instead, b = √(c² − a²). Some triangles have three whole-number sides, called Pythagorean triples, such as 3, 4, 5 and 5, 12, 13. In this simulator you stretch the two short sides, see the squares and areas update live, and switch to a mode that asks you to find the missing side.
Check your understanding
Question 1 of 4
A right triangle has short sides 6 and 8. How long is the hypotenuse?
The hypotenuse is 13 and one short side is 5. What is the other short side?
Does the theorem work in a triangle with angles 50°, 60° and 70°?
Which of these is a Pythagorean triple?
Explain it in your own words
What changed? Why did it happen? Where might you notice this in everyday life?
For parents & teachers
- Measure a door frame diagonal and check it against the theorem.
- Ask why builders use a 3, 4, 5 triangle to make a true right angle.
- Cut paper squares and fit the two smaller ones into the largest.