Quick answer
Given three sides, the angles follow from the law of cosines: cos A = (b² + c² − a²) ÷ 2bc, and the area from Heron's formula. Sides 3, 4 and 5 give angles of 36.87°, 53.13° and 90° and an area of 6.
Enter the three sides. The calculator returns the three angles and the area. Use this sss triangle calculator to check your own figures.
Related tools: 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.
How to use the SSS Triangle Calculator
- Enter side a (above 0).
- Enter side b (above 0).
- Enter side c (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Side a | A number | > 0 |
| Side b | A number | > 0 |
| Side c | A number | > 0 |
Formula and method
Formula
cos A = (b² + c² − a²) ÷ 2bc; area by Heron's formula
IEEE-754 double precision; shown to at most 6 decimal places.
Law of cosines for each angle; triangle inequality enforced.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Side a = 3; Side b = 4; Side c = 5.
| Result | Value |
|---|---|
| Area | 6 |
| Angle A | 36.869898° |
| Angle B | 53.130102° |
| Angle C | 90° |
Example 2
Inputs: Side a = 2; Side b = 2; Side c = 2.
| Result | Value |
|---|---|
| Area | 1.732051 |
| Angle A | 60° |
| Angle B | 60° |
| Angle C | 60° |
The first line is the headline result (area). Below it the sSS Triangle Calculator lists angle A, angle B and angle C, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Steps
cos A = (16 + 25 − 9) ÷ 40 = 0.8, so A = 36.87°. Similarly B = 53.13° and C = 90°. Heron: s = 6, area = √(6 × 3 × 2 × 1) = 6.
Validity
The sides must satisfy the triangle inequality (any two add to more than the third); degenerate triangles are rejected. Use the Triangle calculator for a type description.
How the result changes with side a
Holding every other input at the first worked example, this table changes only side a. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Side a | Area | Angle A | Angle B |
|---|---|---|---|
| 1.5 | 2.480392 | 14.361512° | 41.409622° |
| 3 | 6 | 36.869898° | 53.130102° |
| 6 | 9.921567 | 82.819244° | 41.409622° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Side c: entering "3" shows "These sides cannot form a triangle." beside that field and no result.
Choosing between the SSS Triangle 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 |
|---|---|---|
| [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 |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Triangle Area Calculator](/math/triangle-area-calculator/) | base and height | Area |
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
- Degenerate triangles rejected
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 side c to confirm this.
Frequently asked questions
Can sides 1, 2, 3 form a triangle?
No. 1 + 2 = 3 is not greater than the third side, so the three lengths collapse into a line.
What if the largest angle is over 90°?
The triangle is obtuse.
How is it different from SAS?
SAS gives two sides and an angle.
What does the SSS Triangle Calculator calculate?
Angles and area of a triangle from its three sides. It returns Area, Angle A, Angle B and Angle C.
What formula does the SSS Triangle Calculator use?
cos A = (b² + c² − a²) ÷ 2bc; area by Heron's formula. Law of cosines for each angle; triangle inequality enforced.
What do I need to enter in the sSS Triangle Calculator?
You enter Side a, Side b and Side c.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Degenerate triangles rejected.
What is an example of a sSS Triangle result?
With side a 3, side b 4 and side c 5, the area is 6.
Related calculators
- 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.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Triangle Area Calculator — Area of a triangle from base and height.
- SAS Triangle Calculator — Solve a triangle from two sides and the included angle.
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.