Solve right triangles completely. Find all three sides, both acute angles, all six trigonometric ratios (sin, cos, tan, csc, sec, cot), area, and perimeter. SOH-CAH-TOA reference included.
A right triangle is a triangle containing one angle of exactly 90°. The side opposite this right angle is the hypotenuse — always the longest side. The other two sides are called legs. Trigonometry — the study of relationships between angles and sides — exists largely because of right triangles. Our free right triangle calculator solves any right triangle completely given any valid combination of sides and angles.
SOH-CAH-TOA is the most widely used mnemonic for remembering the three primary trigonometric ratios. For an acute angle A in a right triangle:
| Mnemonic | Ratio | Definition |
|---|---|---|
| SOH | sin A | Opposite / Hypotenuse |
| CAH | cos A | Adjacent / Hypotenuse |
| TOA | tan A | Opposite / Adjacent |
"Opposite" is the side opposite to angle A; "adjacent" is the side next to angle A (not the hypotenuse); "hypotenuse" is always the longest side opposite the right angle. Note that sin A = cos B and cos A = sin B, since A + B = 90° in any right triangle.
Three additional (reciprocal) trig functions complete the set:
| Function | Abbreviation | Definition | Reciprocal of |
|---|---|---|---|
| Cosecant | csc A | Hypotenuse / Opposite | sin A |
| Secant | sec A | Hypotenuse / Adjacent | cos A |
| Cotangent | cot A | Adjacent / Opposite | tan A |
When you know a ratio and want the angle, use the inverse functions: arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹). For example: if sin A = 0.5, then A = arcsin(0.5) = 30°. Our calculator uses these automatically when solving from two known sides.
Sides in ratio 1 : √3 : 2. If hypotenuse = 2, then the side opposite 30° = 1 and the side opposite 60° = √3 ≈ 1.732. These ratios come from bisecting an equilateral triangle. Trig values: sin 30° = 0.5, cos 30° = √3/2 ≈ 0.866, tan 30° = 1/√3 ≈ 0.577.
Sides in ratio 1 : 1 : √2. If legs = 1, hypotenuse = √2 ≈ 1.414. This is an isosceles right triangle. Trig values: sin 45° = cos 45° = √2/2 ≈ 0.707, tan 45° = 1.
Given any two of the five quantities (three sides + two acute angles), you can solve for the rest. Our calculator handles three common combinations:
Navigation uses bearings measured clockwise from north. To find the east-west and north-south components of a journey of distance d at bearing θ: eastward component = d·sin(θ); northward component = d·cos(θ). This decomposition into horizontal and vertical components uses SOH-CAH-TOA with a right triangle formed by the journey vector and the north direction. Maritime navigation (used extensively around the UK's coastline) and aviation both use these trigonometric decompositions constantly.
The angle of elevation is the angle above horizontal to a higher object; the angle of depression is the angle below horizontal to a lower object. If you stand 100 metres from the base of a building and the angle of elevation to the top is 35°, the building height = 100 × tan(35°) ≈ 70.0 metres. This is a direct application of TOA (tan = opposite/adjacent).
In the UK, trigonometry is introduced at GCSE level (SOH-CAH-TOA for right triangles) and extended at A-level Pure Mathematics (Edexcel, AQA, OCR) to include the unit circle, exact values, identities, and the Laws of Sines and Cosines for non-right triangles. Reciprocal functions (csc, sec, cot) appear in A-level Further Mathematics and A-level Mathematics Year 2.
In the US, trigonometry is typically taught in a dedicated Trigonometry or Pre-Calculus course in 10th or 11th grade. SOH-CAH-TOA and right triangle solving appear in Algebra 2 (some curricula), while the full unit circle, radians, and identities are covered in Pre-Calculus. The ACT Mathematics section regularly tests right triangle trigonometry.
Structural engineers use trigonometry to resolve forces into components. A force F applied at angle θ to the horizontal has horizontal component F·cos(θ) and vertical component F·sin(θ). Bridge cable tensions, truss forces, and inclined plane problems all use this decomposition. Electrical engineers use trig functions to analyse AC circuits with phase angles. Surveyors use theodolites to measure angles and then apply trigonometry to calculate distances to inaccessible points.
| To Find | Known | Formula |
|---|---|---|
| Opposite | Angle, Hypotenuse | opp = hyp × sin(A) |
| Adjacent | Angle, Hypotenuse | adj = hyp × cos(A) |
| Opposite | Angle, Adjacent | opp = adj × tan(A) |
| Hypotenuse | Angle, Opposite | hyp = opp / sin(A) |
| Hypotenuse | Angle, Adjacent | hyp = adj / cos(A) |
| Angle A | opp, hyp | A = arcsin(opp/hyp) |
| Angle A | adj, hyp | A = arccos(adj/hyp) |
| Angle A | opp, adj | A = arctan(opp/adj) |
SOH-CAH-TOA is a mnemonic for the three trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It applies to acute angles in right triangles and is the foundation of trigonometry taught in GCSE (UK) and Geometry/Algebra 2 (US).
Use inverse trig functions: if you know two sides, find the angle using arcsin, arccos, or arctan. For example, if opposite = 4 and hypotenuse = 8, then A = arcsin(4/8) = arcsin(0.5) = 30°. The third angle is always 90° - A since all angles sum to 180°.
Sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj, csc A = hyp/opp (reciprocal of sin), sec A = hyp/adj (reciprocal of cos), cot A = adj/opp (reciprocal of tan). All six can be derived from any two sides of a right triangle.
A right triangle with angles 30°, 60°, and 90°. Side ratios are 1:√3:2. If the short leg (opposite 30°) is x, then the long leg (opposite 60°) is x√3 and the hypotenuse is 2x. These exact values are used in UK A-level and US Pre-Calculus.
An isosceles right triangle with two 45° angles. Side ratios are 1:1:√2. If each leg = x, the hypotenuse = x√2. sin 45° = cos 45° = √2/2 ≈ 0.707, tan 45° = 1.
The hypotenuse is the longest side of a right triangle, always opposite the 90° angle. It satisfies the Pythagorean theorem: c = √(a²+b²). In a 3-4-5 right triangle, the hypotenuse is 5. For a right triangle with legs 5 and 12, the hypotenuse is 13.
Use the tangent function: height = distance × tan(angle of elevation). Standing 50 m from a building with angle of elevation 40°: height = 50 × tan(40°) ≈ 50 × 0.839 ≈ 41.95 m. This is a real-world application of TOA from SOH-CAH-TOA.
Degrees divide a full circle into 360 equal parts. Radians measure angles as the ratio of arc length to radius: a full circle = 2π ≈ 6.283 radians. To convert: radians = degrees × π/180. Radians are used in calculus, physics, and programming (most trig functions in code require radians).
Results are based on standard trigonometric formulas and your inputs. For navigation, engineering, or safety-critical calculations, always verify with professional tools and qualified experts.