Quick answer
Given two legs a and b, c = √(a² + b²) and α = arctan(a ÷ b). Legs of 3 and 4 give a hypotenuse of 5, angles of 36.8699° and 53.1301° and an area of 6.
Choose what you know (both legs, or a leg and the hypotenuse) and enter the values. The calculator returns the missing side, both acute angles and the area. Use this right triangle side and angle calculator to check your own figures.
Related tools: Right Triangle Calculator: Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter. Right Triangle Area Calculator: Area of a right triangle from its legs. Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides.
How to use the Right Triangle Side and Angle Calculator
- Choose known sides: Both legs (a, b) and Leg a and hypotenuse c.
- Enter leg a (above 0).
- Enter leg b or hypotenuse c (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Known sides | Choose one: Both legs (a, b); Leg a and hypotenuse c | Starts at Both legs (a, b) |
| Leg a | A number | > 0 |
| Leg b or hypotenuse c | A number | > 0 |
Formula and method
Formula
c² = a² + b²; α = arctan(a ÷ b)
IEEE-754 double precision; shown to at most 6 decimal places.
Pythagorean theorem plus inverse trigonometry.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Known sides = Both legs (a, b); Leg a = 3; Leg b or hypotenuse c = 4.
| Result | Value |
|---|---|
| Hypotenuse c | 5 |
| Angle α | 36.869898° |
| Angle β | 53.130102° |
| Area | 6 |
Example 2
Inputs: Known sides = Leg a and hypotenuse c; Leg a = 5; Leg b or hypotenuse c = 13.
| Result | Value |
|---|---|
| Leg b | 12 |
| Angle α | 22.619865° |
| Angle β | 67.380135° |
| Area | 30 |
The first line is the headline result (hypotenuse c). Below it the right Triangle Side and Angle Calculator lists angle α, angle β and area, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Two input modes
With a leg and the hypotenuse, the other leg is √(c² − a²). Hypotenuse 5 with leg 3 gives leg 4. The hypotenuse must exceed the known leg.
Angles
α is opposite leg a: arctan(3 ÷ 4) = 36.87°, and β = 90° − α = 53.13°. Use the Right Triangle calculator for perimeter and special triangles.
How the result changes with leg a
Holding every other input at the first worked example, this table changes only leg a. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Leg a | Hypotenuse c | Angle α | Angle β |
|---|---|---|---|
| 1.5 | 4.272002 | 20.556045° | 69.443955° |
| 3 | 5 | 36.869898° | 53.130102° |
| 6 | 7.211103 | 56.309932° | 33.690068° |
| 12 | 12.649111 | 71.565051° | 18.434949° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Leg b or hypotenuse c: entering "5" shows "The hypotenuse must be longer than the leg." beside that field and no result.
Choosing between the Right Triangle Side and Angle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Right Triangle Area Calculator](/math/right-triangle-area/) | leg a and leg b | Area |
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [45-45-90 Triangle Calculator](/math/45-45-90-triangle-calculator/) | leg length | Leg |
| [30-60-90 Triangle Calculator](/math/30-60-90-triangle-calculator/) | short leg length | Short leg |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Hypotenuse must exceed the known leg
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for leg b or hypotenuse c to confirm this.
Frequently asked questions
How do I find the angle from two sides?
Use arctan of opposite over adjacent.
What if I know an angle and a side?
Use the SOHCAHTOA calculator.
Do the acute angles add to 90°?
Yes. The right angle is 90°, so the two acute angles sum to 90°: 36.87° + 53.13° in the example.
What does the Right Triangle Side and Angle Calculator calculate?
Complete a right triangle from any two sides. It returns Hypotenuse c, Angle α, Angle β and Area.
What formula does the Right Triangle Side and Angle Calculator use?
c² = a² + b²; α = arctan(a ÷ b). Pythagorean theorem plus inverse trigonometry.
What do I need to enter in the right Triangle Side and Angle Calculator?
You enter Known sides, Leg a and Leg b or hypotenuse c.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Hypotenuse must exceed the known leg.
What does the “Known sides” option change?
It sets how the right Triangle Side and Angle is worked out. The choices are Both legs (a, b) and Leg a and hypotenuse c.
What is an example of a right Triangle Side and Angle result?
With known sides Both legs (a, b), leg a 3 and leg b or hypotenuse c 4, the hypotenuse c is 5.
Related calculators
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Right Triangle Area Calculator — Area of a right triangle from its legs.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- 45-45-90 Triangle Calculator — Sides of a 45°-45°-90° triangle from one leg.
- 30-60-90 Triangle Calculator — Sides of a 30°-60°-90° triangle from the short leg.
- Triangle Area Calculator — Area of a triangle from base and height.
- Triangle Centroid Calculator — Centroid of a triangle from its vertices.
- Polygon Interior Angle Calculator — Interior and exterior angles of a polygon.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.