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.
When to Use This Calculator
- You're solving a geometry or trigonometry homework problem and need to find unknown sides or angles
- You're calculating land area from three surveyed boundary measurements
- You're working on a construction or engineering project requiring precise angle or distance calculations
- You need to verify triangle measurements for a woodworking, architecture, or 3D modeling project
- You're studying for a math exam and want to practice with step-by-step solutions
- You're working on navigation, surveying, or triangulation problems in the field
Steps:
- 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.
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)
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
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
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
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
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
Related Concepts
- Law of Cosines: A formula relating one side of a triangle to the other two sides and the included angle, generalizing the Pythagorean theorem to all triangles
- Law of Sines: A relationship stating that the ratio of each side to the sine of its opposite angle is constant for all three sides
- Heron's Formula: A method for computing triangle area directly from the three side lengths using the semi-perimeter
- Pythagorean Theorem: In a right triangle, the square of the hypotenuse equals the sum of squares of the two legs (c² = a² + b²)
- Triangle Inequality: The rule that the sum of any two side lengths must be greater than the third side for a valid triangle to exist
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.
Interpreting Your Results
The calculator supports three configurations: SSS (three sides known), SAS (two sides and the included angle), and right triangle (two legs known). Each mode uses different mathematical relationships to solve for the remaining values. The step-by-step solution shows you exactly which formula was applied at each stage, making it easy to follow the derivation or reproduce it by hand.
The area calculation uses Heron's formula for SSS mode, which works for any triangle regardless of type. For SAS mode, the area formula uses the sine of the included angle. The right triangle mode uses the simplest approach — half the product of the two legs. All three methods converge to the same area for a given set of measurements, so you can use any mode as a cross-check.
The triangle type indicator tells you whether the triangle is acute (all angles < 90°), right (one angle = 90°), or obtuse (one angle > 90°). This classification is important in engineering because structural load paths differ depending on triangle type, and in trigonometry because different identity sets apply to acute versus obtuse triangles.