Two Values Determine Everything
A right triangle has one angle fixed at 90 degrees, which leaves only two independent values. Give any two — two sides, or a side and an acute angle — and the rest follow from the Pythagorean theorem and basic trigonometry.
SOHCAHTOA
The three ratios, from the perspective of one acute angle:
| Ratio | Definition | Mnemonic |
|---|---|---|
| sine | opposite / hypotenuse | SOH |
| cosine | adjacent / hypotenuse | CAH |
| tangent | opposite / adjacent | TOA |
Because the two acute angles sum to 90 degrees, the sine of one equals the cosine of the other — which is where the "co" in cosine comes from.
Special Right Triangle Chart
| Angles | Side ratio | Exact form |
|---|---|---|
| 45-45-90 | 1 : 1 : 1.414 | 1 : 1 : √2 |
| 30-60-90 | 1 : 1.732 : 2 | 1 : √3 : 2 |
| 3-4-5 | 3 : 4 : 5 | 36.87° and 53.13° |
| 5-12-13 | 5 : 12 : 13 | 22.62° and 67.38° |
The first two are worth memorising because their ratios are exact. They come from halving a square and an equilateral triangle respectively, which is why they appear so often in geometry problems.
Useful Secondary Values
- Area = ½ab, using the two legs. No separate height is needed, since each leg is the height relative to the other.
- Altitude to the hypotenuse = ab/c. It divides the triangle into two smaller triangles similar to the original.
- Circumradius = c/2. The hypotenuse is always a diameter of the circumscribed circle — Thales' theorem.
- Inradius = (a + b − c)/2.
Thales' Theorem
Any triangle inscribed in a semicircle with the diameter as its base is a right triangle. Equivalently, the hypotenuse of a right triangle is always a diameter of its circumscribing circle. It is one of the oldest theorems in geometry and provides a construction for a right angle using only a compass.
Frequently Asked Questions
Can a right triangle be equilateral?
No. Equilateral triangles have three 60-degree angles. A right triangle can be isosceles — the 45-45-90 — but never equilateral.
How do I find an angle from two sides?
Use the inverse function. If you know the opposite and hypotenuse, the angle is arcsin(opposite/hypotenuse). With opposite and adjacent, use arctan.
What if I only know one side?
Not enough. One side leaves infinitely many similar triangles. You need a second side or an angle to fix the shape.