Calculate the angles and area of a triangle from its three side lengths.
Angles are found using the law of cosines; area uses Heron's formula with semi-perimeter s = (a+b+c)/2.
"Solving" a triangle means finding all unknown sides and angles from a known combination of sides and/or angles. Depending on what's given (SSS, SAS, ASA, AAS, or SSA), different trigonometric laws apply — the Law of Sines, the Law of Cosines, or basic angle-sum rules.
Used when you know two sides and the included angle (SAS), or all three sides (SSS) and need to find an angle by rearranging the formula.
Used when you know two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA) — though SSA can sometimes produce two valid triangle solutions (the "ambiguous case").
| Rule | Formula |
|---|---|
| Angle sum | A + B + C = 180° |
| Area (base & height) | Area = ½ × base × height |
| Area (SAS) | Area = ½ab·sin(C) |
| Perimeter | P = a + b + c |
When one angle is exactly 90°, the simpler Pythagorean theorem applies directly: a² + b² = c², where c is the hypotenuse. Right triangles are a special case of the more general laws above, and trigonometric ratios (sine, cosine, tangent) work directly off the two non-right angles.
Using Heron's formula: s = (7+8+9)/2 = 12. Area = √[12(12−7)(12−8)(12−9)] = √(12×5×4×3) = √720 ≈ 26.83. Using the Law of Cosines to find angle A (opposite side 7): cos(A) = (8²+9²−7²)/(2×8×9) = 96/144 = 0.667, so A ≈ 48.19°. The remaining angles follow similarly, summing to 180°.
Why does SSA sometimes give two possible triangles? Because the sine function is positive for both an angle and its supplement, so two different triangle shapes can satisfy the same given measurements.
When can't a triangle be formed? When the given three sides don't satisfy the triangle inequality (each side must be shorter than the sum of the other two), no valid triangle exists.
Is the Law of Cosines just a generalized Pythagorean theorem? Yes — when the included angle is exactly 90°, cos(90°)=0, and the Law of Cosines reduces exactly to the Pythagorean theorem.
What's Heron's formula used for? It calculates a triangle's area directly from its three side lengths alone, without needing to know any angle or height first — useful whenever only SSS data is available.
How do I know which law to use — Sines or Cosines? Use the Law of Cosines when you have three sides (SSS) or two sides plus the included angle (SAS); use the Law of Sines when you have two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA).
What's the difference between an included and non-included angle? An included angle sits directly between the two known sides (as in SAS); a non-included angle is opposite one of the two known sides instead (as in SSA), which is why SSA can produce an ambiguous result.
Can a triangle have more than one right angle? No — since all three angles must sum to exactly 180°, having two 90° angles would leave no room for a third positive angle, so at most one angle can be 90° (or more, obtuse) in any valid triangle.
For a triangle with sides 5, 6, and 7, Heron's formula gives a semi-perimeter of 9, and an area of √(9×4×3×2) = √216 ≈ 14.7 square units — the three angles can then be found individually using the law of cosines.
Law of cosines: A formula relating a triangle's side lengths to the cosine of one of its angles.
Heron's formula: A method for finding a triangle's area directly from its three side lengths.
Triangle inequality: The rule that the sum of any two sides of a triangle must exceed the third side.