How to Solve a Right Triangle

A right triangle has one 90° angle, and that single fixed corner is what makes it so predictable to work with. To solve a triangle means to pin down every measurement in it: all three side lengths and all three angles. The good news is that in a right triangle you rarely need to start with much. Two extra facts on top of the right angle are almost always enough to recover everything else.

The three sides, named from an angle

Trigonometry does not give the sides fixed names like "left" or "bottom." Instead it labels them relative to whichever acute angle you are focused on. The hypotenuse is the longest side, sitting directly across from the right angle; it never changes no matter which angle you pick. The opposite side is the one facing your chosen angle across the triangle, and the adjacent side is the one that forms the angle together with the hypotenuse. Shift your attention to the other acute angle and the opposite and adjacent sides trade roles, while the hypotenuse stays exactly where it is.

Finding a missing side

Three ratios connect an angle to a pair of its sides:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent

The trick is to choose the ratio that mentions the side you already know together with the side you want. If you know the hypotenuse and want the opposite side, sine is your tool. If you know the adjacent side and want the opposite one, tangent connects them. Once the right ratio is chosen, a single multiplication or division isolates the unknown length.

Finding a missing angle

When you know two sides but no acute angle, run the ratios in reverse using the inverse functions sin⁻¹, cos⁻¹, and tan⁻¹. For instance, if the opposite side is 3 and the adjacent side is 4, then tan θ = 3/4 = 0.75, so θ = tan⁻¹(0.75) ≈ 36.87°. The inverse function simply asks the question "which angle has this ratio?"

Two rules that tie it together

The Pythagorean theorem, a² + b² = c² with c the hypotenuse, lets you find any side from the other two without touching an angle at all. The angle-sum rule says the three interior angles of any triangle total 180°. Because one of them is already 90°, the two acute angles must add up to 90° on their own, which means each is the complement of the other. Together these two facts often let you finish a triangle with pure arithmetic once trig has given you a foothold.

Worked example

Suppose one acute angle measures 35° and the hypotenuse is 12 units long. Find the remaining two sides and the third angle.

1. Third angle. 180° − 90° − 35° = 55°.
2. Opposite side. Since sin 35° ≈ 0.5736, the opposite side is 12 × 0.5736 ≈ 6.88.
3. Adjacent side. Since cos 35° ≈ 0.8192, the adjacent side is 12 × 0.8192 ≈ 9.83.

It is always worth checking the result against Pythagoras: 6.88² + 9.83² ≈ 47.3 + 96.6 = 143.9, which rounds neatly to 12² = 144. The small gap is nothing more than rounding along the way, so the triangle is fully solved: sides 6.88, 9.83, and 12, with angles 35°, 55°, and 90°.

Two starting situations

In practice a right-triangle problem lands in one of two shapes. Either you are given an angle plus a side, in which case a trig ratio finds a second side and the angle-sum rule finds the last angle, as in the example above. Or you are given two sides and no acute angle, in which case Pythagoras finds the third side while an inverse trig function recovers an angle. Learning to recognise which situation you are in is half the battle, because it tells you immediately whether to reach for a ratio or for the theorem first.

One practical caution: before trusting any number, make sure your calculator is set to degrees and not radians. A triangle that comes out wildly wrong is very often a mode error rather than a mistake in your method.

Once you have the pattern, the whole process becomes mechanical. Try it yourself with the calculators on SinCosTan and watch the numbers fall into place.