SAT® · Math: Geometry and Trigonometry
The Pythagorean Theorem
For any right triangle, a2 + b2 = c2, where a and b are the legs and c is the hypotenuse, the side across from the right angle. It only works when the triangle actually has a 90 degree angle.
The theorem says the square built on the hypotenuse has the same area as the two squares built on the legs combined. You can see why with a rearrangement proof: take a square with side a+b, and place four copies of the same right triangle inside it, pointing in a pinwheel pattern. The leftover space in the middle is a tilted square with side c, so its area is c2. Slide the triangles around so they pair up into two rectangles instead, and the leftover space splits into two squares, with areas a2 and b2. Same four triangles, same big square, so the leftover area has to match: a2 + b2 = c2.
To use it, identify the right angle first. The hypotenuse is always the side opposite that angle, and it is always the longest side. If legs are 9 and 12, the hypotenuse is √(92+122) = √(81+144) = √225 = 15. If the hypotenuse is 26 and one leg is 10, the other leg is √(262-102) = √(676-100) = √576 = 24. The theorem breaks the moment the angle stops being exactly 90 degrees, which is why it cannot be applied directly to obtuse or acute triangles.
The idea extends naturally into solids. For a box with dimensions l, w, and h, the diagonal across the base is √(l2+w2), and running the theorem again up the height gives the space diagonal: √(l2+w2+h2). This connects directly to distance formulas and to right-triangle trig, since sine, cosine, and tangent all describe ratios between the same three sides the theorem relates.
Key terms
- Hypotenuse
- The longest side of a right triangle, located opposite the right angle.
- Leg
- Either of the two shorter sides of a right triangle that form the right angle.
- Space diagonal
- The line segment connecting two opposite corners of a three-dimensional box, passing through its interior.
- Obtuse angle
- An angle greater than 90 degrees, which makes the side opposite it longer than the right-triangle case.
Watch it explained
- How it worksHow does rearranging four copies of a right triangle inside a square prove a squared plus b squared equals c squared?
- Worked exampleFind the hypotenuse of a right triangle whose legs measure 9 and 12.
- Worked exampleFind the missing leg of a right triangle with hypotenuse 26 and one leg of 10.
- Common mistakeWhy doesn't the leg-squared-plus-leg-squared rule work on a triangle containing a 100 degree angle?
- Going furtherWhat happens when you push the theorem into three dimensions to find the long diagonal of a box?
Check yourself
End of topic.