Pythagorean Theorem
In a right triangle, a² + b² = c².
About this topic
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². A 3-4-5 triangle satisfies it since 9 + 16 = 25.
Use it to find a missing side, check whether a triangle is right-angled, or compute distances on a grid. It has hundreds of proofs and is used in construction, navigation and the distance formula.
Worked example: a ladder leans on a wall with its foot 3 m from the wall and the top 4 m up. Its length is √(3² + 4²) = 5 m. To find a leg, if the hypotenuse is 13 and one leg is 5, then the other leg is √(169 − 25) = 12. To test whether sides 6, 7 and 9 form a right triangle, 36 + 49 = 85 differs from 81, so it is not right, and it is acute since 85 is greater than 81.
Carpenters check corners with it, and phone and GPS software use it for distance. The most common mistake is putting the hypotenuse in the wrong place: c is always the longest side, opposite the right angle. Another is adding when finding a leg instead of subtracting. Remember to take the square root at the end, and check that the hypotenuse is the largest side.
Key ideas
- c is the hypotenuse opposite the right angle
- Used to find a missing side
- Converse tests for right triangles
Quick check
Use cases: where Pythagorean Theorem helps
- Homework: review the key idea behind Pythagorean Theorem before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Pythagorean Theorem without notes.
- Visual learning: play with the interactive lab to see how Pythagorean Theorem changes when you adjust its inputs.
- Teaching a friend: explain Pythagorean Theorem in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Pythagorean Theorem?
In a right triangle, a² + b² = c².
What will I learn in Pythagorean Theorem?
c is the hypotenuse opposite the right angle. Used to find a missing side. Converse tests for right triangles.
Is Pythagorean Theorem part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Pythagorean Theorem lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Pythagorean Theorem?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.