Choose tangent ratio, opposite side length, or adjacent side length, then enter the two known values. Use positive side lengths in the same unit and a positive, unitless tangent ratio.
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Tangent Ratio Formula
The tangent ratio compares the opposite side and adjacent side of a right triangle for a given acute angle.
- T = tangent ratio
- O = opposite side length
- A = adjacent side length
The calculator uses the first formula when you enter the opposite and adjacent side lengths. It divides the opposite side by the adjacent side to find the tangent ratio.
For the opposite-side mode, it uses O = A * T. For the adjacent-side mode, it uses A = O / T. Both side lengths and the tangent ratio must be positive for a nondegenerate right triangle with an acute reference angle.
Common Tangent Ratios for Special Angles
These values can help you check whether a tangent ratio is reasonable for common right-triangle angles.
| Angle | Exact Tangent Ratio | Decimal Approximation |
|---|---|---|
| 30° | 1 / √3 | 0.5774 |
| 45° | 1 | 1.0000 |
| 60° | √3 | 1.7321 |
Tangent Ratio Result Guide
| Tangent Ratio | What It Means |
|---|---|
| Less than 1 | The opposite side is shorter than the adjacent side. |
| Equal to 1 | The opposite and adjacent sides are the same length. |
| Greater than 1 | The opposite side is longer than the adjacent side. |
Example Problems
Example 1: Find the tangent ratio
You have a right triangle with an opposite side of 9 and an adjacent side of 12.
The tangent ratio is 0.7500.
Example 2: Find the opposite side
You know the adjacent side is 8 and the tangent ratio is 1.5.
The opposite side length is 12.0000.
FAQ
What is the tangent ratio?
The tangent ratio is the opposite side divided by the adjacent side in a right triangle. It is written as tan(θ) = opposite / adjacent, where θ is the angle you are measuring from.
Which sides are opposite and adjacent?
The opposite side is across from the angle you are using. The adjacent side is next to that angle and is not the hypotenuse. The hypotenuse is not used in the tangent ratio.
Can the tangent ratio be greater than 1?
Yes. A tangent ratio greater than 1 means the opposite side is longer than the adjacent side. For example, if the opposite side is 10 and the adjacent side is 5, the tangent ratio is 2.
