Triangle Calculator
Find the missing sides, angles, area and perimeter of any triangle.
Solving a triangle
A triangle has three sides and three angles, and three well-chosen values fix the rest. This calculator handles the three most common cases: all three sides, two sides with the angle between them, and two angles with the side between them.
The rules it uses
- Law of cosines: c² = a² + b² − 2ab cos C, used to find a side or an angle.
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C.
- Heron’s formula for the area: √(s(s − a)(s − b)(s − c)), with s = (a + b + c) ÷ 2.
- The three angles always add up to 180°.
Example
On the first tab (three sides (sss)) with the starting values shown (Side a 7; Side b 8; Side c 9), Area comes out as 26.832816. Other results include perimeter: 24; side a: 7; side b: 8.
The calculator also classifies the triangle by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse), and gives the inradius and circumradius.
Which information do you need?
- SSS: all three sides. The calculator uses the law of cosines to find the angles.
- SAS: two sides and the angle between them. It finds the third side with the law of cosines.
- ASA (or AAS): two angles and a side. The third angle is 180° minus the other two, and the law of sines gives the sides.
The key formulas
- Angles add up to 180°.
- Law of cosines: c² = a² + b² − 2ab cos C.
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C.
- Area from three sides (Heron): √(s(s − a)(s − b)(s − c)), where s is half the perimeter.
Example: a triangle with sides 7, 8 and 9
The half-perimeter is 12, so the area is √(12 × 5 × 4 × 3) = √720 ≈ 26.83. The angles are about 48.19°, 58.41° and 73.40°, which add up to 180°. All angles are under 90°, so it is an acute scalene triangle.
Types of triangle
| Type | Rule |
|---|---|
| Equilateral | All three sides equal, all angles 60° |
| Isosceles | Two sides equal |
| Scalene | No sides equal |
| Right | One angle of 90° |
| Acute | All angles under 90° |
| Obtuse | One angle over 90° |
Three sides form a triangle only if each is shorter than the sum of the other two.
Next steps: Pythagorean Theorem Calculator, Area Calculator, Right Triangle Calculator.
Frequently asked questions
How do I find the angles of a triangle from its sides?
Use the law of cosines: cos A = (b² + c² − a²) ÷ 2bc, and the same pattern for the other angles.
How do I find the area of a triangle with three sides?
Use Heron's formula: the square root of s(s − a)(s − b)(s − c), where s is half the perimeter.
Do the angles of a triangle always add up to 180°?
Yes, in flat (Euclidean) geometry.
What is the difference between SSS, SAS and ASA?
They describe what you know: three sides, two sides with the angle between them, or two angles with the side between them.
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