Triangle calculator
Enter three side lengths to calculate triangle area, perimeter, and interior angles.
What this calculator covers
Use this triangle calculator for side-side-side problems where the side lengths are known but the area or angles are not.
The output keeps Heron's formula and the law of cosines visible, so you can trace how the same three sides determine both the surface area and the interior angles.
Frequently asked questions
- Why do the three side lengths need to satisfy the triangle inequality?
- Any valid triangle requires that the sum of any two sides must exceed the third side. If that condition fails — for example, sides 1, 2, and 10 — no triangle can be formed, so the calculator returns an error rather than producing a meaningless result.
- What formula is used to find area from three side lengths?
- The calculator uses Heron's formula: area equals the square root of s(s−a)(s−b)(s−c), where s is the semiperimeter (half the perimeter). This approach works without needing a height measurement.
- How are the interior angles calculated?
- Each angle is found using the law of cosines, which relates the three side lengths to the cosine of each angle. The third angle is derived by subtracting the first two from 180 degrees to avoid rounding accumulation.
- What units do the sides and area use?
- The calculator is unitless — enter sides in any consistent unit (centimeters, inches, meters) and the area comes back in the square of that same unit. Mixing units will produce incorrect results.
Tool
Run the calculation
Result
RESULT · TRIANGLE AREA
â„–113
Primary result
6 sq units
A triangle with sides 3, 4, and 5 has area 6, perimeter 12, and angles 36.87 deg, 53.13 deg, and 90 deg.
- Perimeter
- 12
- Area
- 6
- Angles
- 36.87 deg, 53.13 deg, 90 deg
Step-by-step solution
- 1.Add the three sides to get the perimeter 12, then halve it to get the semiperimeter 6.
- 2.Use Heron's formula sqrt(s(s-a)(s-b)(s-c)) to find the area 6.
- 3.Apply the law of cosines to convert the three side lengths into interior angles of 36.87 deg, 53.13 deg, and 90 deg.
Walkthrough
Visual walkthrough
With side-side-side input, a triangle is fully determined, so one pass through Heron and the law of cosines reveals the rest.
01
Add the side lengths
3 + 4 + 5 = 12
Perimeter and semiperimeter are the launch point for Heron's area formula.
02
Use Heron's formula
sqrt(s(s-a)(s-b)(s-c)) = 6
Heron's formula turns the three side lengths into area without needing a height measurement.
03
Recover the interior angles
The law of cosines converts side lengths into the angle openings that complete the triangle description.
36.87 deg / 53.13 deg / 90 deg