The short answer
A right triangle has one 90° angle, with the hypotenuse always opposite it (and always the longest side). Trigonometry connects the acute angles to the side ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH-CAH-TOA). Given any two sides, or one side and one acute angle, you can solve for everything else.
Key takeaways
- The two acute angles in a right triangle always add up to 90° — find one and the other follows instantly by subtraction.
- SOH-CAH-TOA is the tool for solving a triangle when you know an angle plus one side, not two sides — that case calls for the Pythagorean theorem instead.
- Area of a right triangle is simply half the product of the two legs: A = ½ab, since the legs themselves already form a right angle.
- A triangle can never have two right angles — the three angles must sum to 180°, leaving no room for a second 90°.
SOH-CAH-TOA at a glance
| Ratio | Formula | Mnemonic |
|---|---|---|
| sin θ | opposite / hypotenuse | SOH |
| cos θ | adjacent / hypotenuse | CAH |
| tan θ | opposite / adjacent | TOA |
Worked example: solving from an angle and a side
Angle A = 30°, hypotenuse c = 10. Find both legs and the remaining angle.
a = c × sin(30°) = 10 × 0.5 = 5
b = c × cos(30°) = 10 × 0.866 ≈ 8.66
Angle B = 90° − 30° = 60°
Check with the Pythagorean theorem: 5² + 8.66² ≈ 25 + 75 = 100 = 10² — confirming the trigonometric result is consistent with the triangle's sides.
Area and perimeter formulas
Area = ½ × a × b = ½ × 5 × 8.66 ≈ 21.65
Perimeter = a + b + c = 5 + 8.66 + 10 = 23.66
Because the two legs of a right triangle already meet at a 90° angle, they double as base and height directly — no separate height calculation needed, unlike with other triangle types.
Common mistakes to avoid
- Using tangent when the hypotenuse is involved — tan only relates the two legs to each other, not to the hypotenuse.
- Mixing up "opposite" and "adjacent" — these labels depend on which acute angle you're using, even though the hypotenuse itself never changes.
- Forgetting to check degree vs. radian mode before computing sin, cos, or tan by hand or with a separate calculator.
- Searching for a separate "height" for the area formula — in a right triangle, the two legs already serve as base and height.
Related calculators
- Pythagorean Theorem Calculator — solve for a missing side when you know the other two sides directly.
- Triangle Calculator — work with triangles that aren't right-angled.
- Distance Calculator — see the same right-triangle logic applied to coordinate points.
- Scientific Calculator — compute sine, cosine, and tangent directly for any angle.