Triangle Calculator

Solve any triangle instantly. Enter three sides (SSS), two sides and an angle (SAS), or two legs of a right triangle to find all angles, sides, area, and perimeter with step-by-step solutions.

What is Triangle Calculator?

A triangle is fully determined once you know enough of its sides and angles — this calculator supports the three most common cases. SSS (side-side-side) solves a triangle from all three side lengths using the Law of Cosines and Heron's Formula. SAS (side-angle-side) solves a triangle from two sides and the angle between them. The right-triangle mode uses the Pythagorean theorem to find the hypotenuse and both acute angles from the two legs.

Whichever mode you use, the calculator returns every missing side and angle, the area, the perimeter, and a complete step-by-step solution so you can see exactly how each value was derived.

Formula

Law of Cosines: c² = a² + b² - 2ab·cos(C)

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Heron's Formula: Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2

Pythagorean Theorem: c² = a² + b² (right triangles only)

SAS Area: Area = ½ab·sin(C)

How to Calculate

Choose the mode that matches what you know: three sides (SSS), two sides and the included angle (SAS), or the two legs of a right triangle.
Enter the known measurements.
The calculator solves for every missing side, angle, the area, and the perimeter.
Review the full step-by-step derivation for each value.

Example

For a triangle with sides a=5, b=6, c=7 (SSS mode): angle A ≈ 44.4°, angle B ≈ 57.1°, angle C ≈ 78.5°, and the area (via Heron's formula) is √(9×4×3×2) = √216 ≈ 14.7.

Key Benefits

Solves three different triangle configurations (SSS, SAS, right triangle)
Returns all sides, angles, area, and perimeter at once
Full step-by-step working using the Law of Cosines, Law of Sines, and Heron's formula
Automatically flags whether the triangle is right or oblique

Common Mistakes to Avoid

Confusing degrees and radians when entering or interpreting angles
Entering three sides that violate the triangle inequality (no valid triangle exists)
Mixing up which angle is "included" between two sides in SAS mode
Forgetting that angle sum must equal 180° when checking your own work

Pro Tips

In SSS mode, the largest angle is always opposite the longest side
For a right triangle, you can skip the Law of Cosines entirely and just use the Pythagorean theorem
Heron's formula works for any triangle, not just right triangles — it's often faster than trigonometry when you only know the sides
Double-check your triangle is valid: the sum of any two sides must exceed the third

Key Terms Explained

Law of Cosines: Relates one side to the other two sides and the angle between them
Law of Sines: Relates each side to the sine of its opposite angle
Heron's Formula: Computes triangle area directly from the three side lengths
Hypotenuse: The longest side of a right triangle, opposite the right angle

Common Use Cases

Solving geometry and trigonometry homework
Calculating land area from surveyed triangle measurements
Engineering and construction — finding unknown angles or distances
Navigation and triangulation problems

Frequently Asked Questions

What information do I need to solve a triangle?

You need at least three pieces of information, including at least one side length: three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA/AAS), or the two legs of a right triangle.

How is triangle area calculated from three sides?

Heron's formula computes area from the three side lengths alone: Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter, (a+b+c)/2.

What's the difference between the Law of Sines and the Law of Cosines?

The Law of Cosines relates all three sides to one angle and is used when you know SSS or SAS. The Law of Sines relates sides to the sines of their opposite angles and is typically used once you already know one side-angle pair.

Why doesn't my triangle solve — I entered three sides?

The three sides must satisfy the Triangle Inequality: each side must be shorter than the sum of the other two. If a + b ≤ c (or any similar combination), no triangle can be formed with those measurements.

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