What this calculator does
This trigonometry calculator solves a right-angled triangle from the smallest amount of information needed: one angle (other than the right angle) and the length of one side. Every other side, the remaining angle and the triangle's area come out from there.
The three relationships behind it, sine, cosine and tangent, connect an angle to the ratio between two sides of a right triangle. Given which side is known, hypotenuse, opposite or adjacent, the calculator picks the matching ratio and works the other two sides out directly.
The formula
Sine of an angle is the opposite side divided by the hypotenuse; cosine is the adjacent side divided by the hypotenuse; tangent is the opposite side divided by the adjacent side. Given one angle and any one side, rearranging whichever ratio includes that known side gives the other two.
| Term | Meaning |
|---|---|
| Hypotenuse | The longest side of a right triangle, opposite the right angle. |
| Opposite side | The side directly across from the angle being used in the calculation. |
| Adjacent side | The side next to the angle being used, other than the hypotenuse. |
The inputs explained
| Field | What to enter |
|---|---|
| Known angle (not the right angle) (°) | The known angle, in degrees, not counting the right angle itself. |
| Known side | Which side's length you already know. |
| Length of the known side | The length of that known side. |
When to use it
Finding a height using an angle of elevation
A surveyor or hiker measuring the angle up to the top of a structure, plus the known distance to its base, can work out its height without climbing it, using tangent.
Checking a ramp, roof or staircase angle
Given a horizontal run and a target angle, this works out the vertical rise, or given the rise and the slope length, the angle itself.
Solving a right triangle for schoolwork
A GCSE or introductory trigonometry problem that gives one angle and one side is the exact case this calculator is built for, showing every remaining side and angle at once.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the sides change as the known angle increases, for a fixed hypotenuse?
A right triangle with hypotenuse 10, across a range of known angles.
| Known angle | Opposite side | Adjacent side | Other angle (B) |
|---|---|---|---|
| 10° | 1.736 | 9.848 | 80.00° |
| 20° | 3.420 | 9.397 | 70.00° |
| 30° | 5.000 | 8.660 | 60.00° |
| 45° | 7.071 | 7.071 | 45.00° |
| 60° | 8.660 | 5.000 | 30.00° |
| 80° | 9.848 | 1.736 | 10.00° |
Solving from a known opposite side instead of the hypotenuse
A right triangle with a known opposite side of 8, across a range of known angles.
| Known angle | Hypotenuse | Adjacent side |
|---|---|---|
| 15° | 30.910 | 29.856 |
| 30° | 16.000 | 13.856 |
| 45° | 11.314 | 8.000 |
| 60° | 9.238 | 4.619 |
| 75° | 8.282 | 2.144 |
Questions
What are sine, cosine and tangent?
They are the three basic trigonometric ratios in a right triangle: sine is opposite ÷ hypotenuse, cosine is adjacent ÷ hypotenuse, and tangent is opposite ÷ adjacent. Every right-triangle trigonometry problem is solved by picking the ratio that connects the known values to the unknown one.
Why does this calculator need the right angle to not be the known angle?
The right angle is always 90° by definition, so entering it would supply no new information. The calculator needs one of the other two angles, which vary depending on the triangle's shape.
Can I find an angle instead of a side with this calculator?
This page solves sides and the second angle from one known angle and one known side. To find an angle from two known sides instead, see the trigonometric functions calculator or the triangle solver, which works from two sides and the angle between them.
What is the area of a right triangle?
Half the product of the two sides that meet at the right angle, the opposite and adjacent sides here: Area = ½ × opposite × adjacent.
For sine, cosine and tangent of a single angle on their own, see the trigonometric functions calculator. For any triangle from two sides and the angle between them, not just a right triangle, use the triangle solver.