Geometry & Measurement · Grades 8–10
How to Use the Pythagorean Theorem (With Worked Examples)
Know which side is which
The hypotenuse (c) is the side opposite the right angle and is always the longest. The other two sides, a and b, are the legs. Labelling these correctly is more than half the battle.
Finding the hypotenuse: add the squares
Square both legs, add them, then take the square root. If the legs are 5 and 7, the hypotenuse is not a whole number, and that is fine.
Legs 5 and 7
- Area of the triangle: 17.5
- Perimeter: 20.6023
Step-by-step working 4 steps
- 1The theorem
a² + b² = c²
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
- 2Square the legs and add
5² + 7² = 25 + 49 = 74
- 3Take the square root
c = √74 = 8.602325
- 4Check
5² + 7² = 74 and 8.6023² = 74
Finding a leg: subtract the squares
When you know the hypotenuse and one leg, rearrange to a² = c² − b². With c = 13 and b = 12 the missing leg is 5.
Hypotenuse 13, leg 12
- Area of the triangle: 30
- Perimeter: 30
Step-by-step working 4 steps
- 1The theorem
a² + b² = c²
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
- 2Rearrange for the missing leg
a² = c² − b² a² = 13² − 12² = 169 − 144 = 25
- 3Take the square root
a = √25 = 5
- 4Check
5² + 12² = 169 and 13² = 169
Testing for a right angle
If three side lengths satisfy a² + b² = c² (with c the longest), the triangle has a right angle. This is how builders check corners with a 3-4-5 loop of string.
Is 6-8-10 a right triangle?
Step-by-step working 3 steps
- 1The theorem
a² + b² = c²
a and b are the two shorter sides (legs); c is the hypotenuse, opposite the right angle.
- 2Test the sides
a² + b² = 6² + 8² = 100 c² = 10² = 100
- 3Compare
Both sides are equal, so the sides satisfy the theorem.
Common mistakes
Adding when you should subtract
Adding is for finding c. For a leg, subtract the other leg’s square from c².
Stopping at c²
a² + b² equals c², so the last step is a square root.
Applying it to any triangle
It needs a 90° angle. Other triangles use the law of cosines.
Frequently asked questions
Can the theorem be used in 3D?
Yes. Apply it twice: first across the base to get the diagonal, then with the height to get the space diagonal.
What is a Pythagorean triple?
Three whole numbers that satisfy the theorem, such as 3-4-5 or 5-12-13.
