Right triangle calculator for sides, angles and area
Choose the known side, enter its length and one acute angle. HitEquals solves both legs, the hypotenuse, complementary angle, area and perimeter.
Inputs used by this calculator
Right Triangle Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Known side | Hypotenuse c | Leg a — opposite angle A · Leg b — adjacent to angle A · Hypotenuse c | Supplies the known side quantity used by the published equation or result breakdown. |
| Known side length | 10 | 1e-9 to 1000000000000 | Supplies the known side length quantity used by the published equation or result breakdown. |
| Acute angle A (degrees) | 30 | 0.000001 to 89.999999 | Supplies the acute angle a (degrees) quantity used by the published equation or result breakdown. |
Solve a right triangle
A right triangle already has a 90° angle. One additional acute angle and one side determine the remaining sides and angle.
Sine, cosine and tangent
Relative to angle A, a is opposite, b is adjacent and c is the hypotenuse.
sin A=a/ccos A=b/ctan A=a/bComplementary angle
The two acute angles sum to 90°, so B=90°−A. Angles near either boundary can make one leg extremely small or large relative to the other.
Special right triangles
A 45-45-90 triangle has side ratio 1:1:√2. A 30-60-90 triangle has short-leg, long-leg and hypotenuse ratio 1:√3:2.
Area and perimeter
The perpendicular legs form base and height, so area is ab/2. Perimeter is a+b+c.
Worked 30-60-90 triangle
Choose hypotenuse c, enter 10 and angle A=30°. Sine gives a=10×0.5=5. Cosine gives b=10×√3/2≈8.6603, while the complementary angle B is 60°.
- Identify the hypotenuse as the side opposite 90°.
- Match opposite and adjacent legs to angle A.
- Apply sine and cosine without rounding intermediate values.
- Check a²+b²=c².
Measurement precision in real projects
A calculated side can only be as reliable as the measured side and angle. Small angular errors can create large length differences for long, shallow triangles, so survey and construction work should retain guard digits and follow the required field standard.
Special triangle ratios
| Angles | Ratio |
|---|---|
| 45°-45°-90° | 1 : 1 : √2 |
| 30°-60°-90° | 1 : √3 : 2 |
Which trigonometric ratio to use
| Ratio | Relationship | Useful pair |
|---|---|---|
| Sine | sin A=a/c | opposite and hypotenuse |
| Cosine | cos A=b/c | adjacent and hypotenuse |
| Tangent | tan A=a/b | opposite and adjacent |
Frequently asked questions
What information solves a right triangle?
One acute angle and one side are sufficient because the right angle is known.
Which side is the hypotenuse?
It is opposite the 90° angle and is the longest side.
Do acute angles add to 90 degrees?
Yes, because all triangle angles total 180°.
How is triangle area calculated?
Multiply the two perpendicular legs and divide by two.
Can I enter 0 or 90 degrees?
No. Those boundary values do not form a nondegenerate right triangle.
How can I check the solved sides?
Substitute them into a²+b²=c²; the squared legs should equal the squared hypotenuse within rounding.
Why do shallow angles amplify error?
Tangent changes rapidly relative to the small opposite side, so slight angle or length errors can materially move the result.
What to keep in mind
This solves Euclidean right triangles. The entered angle must be greater than 0° and less than 90°.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.