Pythagorean Theorem
In a right triangle the squares on the legs sum to the square on the hypotenuse.
Key idea
In a right triangle the squares on the legs sum to the square on the hypotenuse.
In depth
In a right triangle, a² + b² = c², where c is the hypotenuse opposite the right angle. It says the area of the square on the hypotenuse equals the sum of the areas on the other two sides. Many proofs exist by rearranging areas.
Worked example
Legs 6 and 8: c² = 36 + 64 = 100, so c = 10.
Formula
Real-world uses
- Checking a rectangle is square with diagonals
- Finding the distance across a field or a ladder’s reach
Common mistake
Applying it to a triangle without a right angle, or treating a leg as the hypotenuse.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Pythagorean Theorem?
In a right triangle the squares on the legs sum to the square on the hypotenuse. It is a topic at the intermediate level in Geometry.
What is the formula for Pythagorean Theorem?
The key formula is a² + b² = c².
Where is Pythagorean Theorem used in real life?
Real-world examples: Checking a rectangle is square with diagonals; Finding the distance across a field or a ladder’s reach.
What should I learn before Pythagorean Theorem?
Make sure you are comfortable with Similar Triangles, the previous topic in Geometry.
What is a common mistake with Pythagorean Theorem?
Applying it to a triangle without a right angle, or treating a leg as the hypotenuse.
How can I practise Pythagorean Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Pythagorean Theorem?
Move on to Pythagorean Triples next.
Related intermediate topics in Geometry
- Vertical Angles – Opposite angles formed by intersecting lines are equal.
- Parallel Lines and Transversals – Corresponding and alternate interior angles are equal.
- Exterior Angle Theorem – An exterior angle equals the sum of the two remote interior angles.
- Triangle Inequality – Any two sides sum to more than the third.
- Congruent Triangles – Same shape and size: SSS, SAS, ASA, AAS, HL.
- Similar Triangles – Same shape, proportional sides: AA, SAS~, SSS~.