The formula
a² + b² = c² · tan A = a ÷ b · area = ab ÷ 2
Where it comes from
A right triangle has one angle fixed at 90°, so two more values determine it entirely. Two sides give you the third by Pythagoras and the angles by the inverse ratios; a side and an angle give you everything by the ratios directly.
How to work it out by hand
- Fill in any two values and leave the rest at zero
- Missing sides come from Pythagoras
- Missing angles come from the inverse ratios
- The two acute angles always add to 90°
What is worth knowing
The height on the hypotenuse is the one value people forget, and it is worth knowing because it is what the geometric mean theorem is about: that height is the geometric mean of the two segments it cuts the hypotenuse into. It also gives you the triangle's area a second way, which makes a useful cross-check. One caution when working from an angle: the two acute angles are complementary, so an error in one propagates directly into the other, and a slip of half a degree can move a long side noticeably.