Geometry & Measurement

Triangle Calculator (Three Sides)

Enter the three sides. The calculator first checks the triangle inequality, then finds the perimeter, the area with Heron’s formula, each angle with the law of cosines, and classifies the triangle.

Updated October 3, 2026Written and reviewed by Mateuss M.Method: how we calculate

Try an example:

AnswerArea 6 · Perimeter 12

scalene, right
Angles 36.87°, 53.13°, 90°

Step-by-step working 5 steps

1
Check the triangle inequality
```
Shortest two: 3 + 4 = 7  vs longest: 5
```
The two shorter sides add to more than the longest, so a triangle exists.
2
Perimeter and semi-perimeter
```
P = 3 + 4 + 5 = 12
s = P ÷ 2 = 6
```

Area by Heron’s formula

Area = √(s(s−a)(s−b)(s−c))
= √(6 × 3 × 2 × 1) = 6

Angles by the law of cosines

cos A = (b² + c² − a²) / 2bc  →  A = 36.8699°
B = 53.1301°
C = 90°
Sum = 180°

5
Classify
By sides: scalene. By angles: right.

How to use the triangle calculator (three sides)

Enter the three side lengths in the same unit.
Check the existence test in step 1.
Read the area, perimeter and angles.
Use the classification to see whether the triangle is acute, right or obtuse.

Formulas used

Triangle exists if the two shorter sides sum to more than the longest
s = (a + b + c) / 2
Area = √(s(s−a)(s−b)(s−c))
cos A = (b² + c² − a²) / 2bc

Angles in a triangle always sum to 180°.

Worked examples

Each example below is generated by the same calculator you used above, so the working always matches the answer.

3, 4, 5

AnswerArea 6 · Perimeter 12

scalene, right
Angles 36.87°, 53.13°, 90°

Step-by-step working 5 steps

1
Check the triangle inequality
```
Shortest two: 3 + 4 = 7  vs longest: 5
```
The two shorter sides add to more than the longest, so a triangle exists.
2
Perimeter and semi-perimeter
```
P = 3 + 4 + 5 = 12
s = P ÷ 2 = 6
```

Area by Heron’s formula

Area = √(s(s−a)(s−b)(s−c))
= √(6 × 3 × 2 × 1) = 6

Angles by the law of cosines

cos A = (b² + c² − a²) / 2bc  →  A = 36.8699°
B = 53.1301°
C = 90°
Sum = 180°

5
Classify
By sides: scalene. By angles: right.

7, 8, 9

AnswerArea 26.8328 · Perimeter 24

scalene, acute
Angles 48.19°, 58.41°, 73.4°

Step-by-step working 5 steps

1
Check the triangle inequality
```
Shortest two: 7 + 8 = 15  vs longest: 9
```
The two shorter sides add to more than the longest, so a triangle exists.
2
Perimeter and semi-perimeter
```
P = 7 + 8 + 9 = 24
s = P ÷ 2 = 12
```

Area by Heron’s formula

Area = √(s(s−a)(s−b)(s−c))
= √(12 × 5 × 4 × 3) = 26.832816

Angles by the law of cosines

cos A = (b² + c² − a²) / 2bc  →  A = 48.1897°
B = 58.4119°
C = 73.3985°
Sum = 180°

5
Classify
By sides: scalene. By angles: acute.

5, 5, 5

AnswerArea 10.8253 · Perimeter 15

equilateral, acute
Angles 60°, 60°, 60°

Step-by-step working 5 steps

1
Check the triangle inequality
```
Shortest two: 5 + 5 = 10  vs longest: 5
```
The two shorter sides add to more than the longest, so a triangle exists.
2
Perimeter and semi-perimeter
```
P = 5 + 5 + 5 = 15
s = P ÷ 2 = 7.5
```

Area by Heron’s formula

Area = √(s(s−a)(s−b)(s−c))
= √(7.5 × 2.5 × 2.5 × 2.5) = 10.825318

Angles by the law of cosines

cos A = (b² + c² − a²) / 2bc  →  A = 60°
B = 60°
C = 60°
Sum = 180°

5
Classify
By sides: equilateral. By angles: acute.

2, 3, 6 (impossible)

These lengths cannot form a triangle: the two shorter sides must add to more than the longest.

Common mistakes to avoid

Skipping the existence check

Sides 2, 3 and 6 cannot form a triangle. Heron’s formula would then fail on a negative number.

Using base × height ÷ 2 without a height

When you only have sides, Heron’s formula is the direct route to the area.

Forgetting that sides set the angles

Three side lengths fix the triangle completely — no other angles are possible.

Frequently asked questions

What is Heron’s formula?

A way to find a triangle’s area from its three sides alone, using the semi-perimeter s.

How is a right triangle detected?

If the largest angle is 90° to within rounding, the triangle is labelled right.