What a Triangle Calculator Does (and When You Need One)
A triangle calculator helps you find missing values in a triangle quickly and accurately. In geometry and trigonometry, triangles show up everywhere: measuring heights, distances, roof pitch, ramps, navigation, engineering drawings, and even algorithms (graphics, physics engines, and simulations). Because a triangle is fully determined by a small set of measurements, a good calculator can “solve” it: compute all unknown sides and angles and then derive helpful measurements like area, perimeter, and special radii.
This tool focuses on the most common and unambiguous solving cases: SSS (three sides), SAS (two sides and the included angle), and ASA/AAS (two angles and one side). It also includes a dedicated right triangle mode for fast Pythagorean and trig calculations.
Triangle Notation (Sides a, b, c and Angles A, B, C)
A standard convention is used in most textbooks and courses: the side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. That pairing matters because the two main triangle laws (Law of Sines and Law of Cosines) relate each side to its opposite angle.
The “angle sum” rule is the first validation check for angle-based inputs: if your angles don’t add up to less than 180°, no triangle exists.
Valid Triangles and the Triangle Inequality
Not every set of three numbers can form a triangle. For side lengths, the triangle inequality must hold: each side must be shorter than the sum of the other two sides. If a side equals the sum of the other two, you get a degenerate triangle (flat line) with area 0.
The calculator enforces these checks in SSS and Heron modes. If the inequality fails, the result is shown as an error so you can correct your inputs immediately.
How Triangles Are Solved: The Three Core Cases
A triangle is determined when you know enough independent information. In practice, that means:
| Case | What you know | Main formulas used | Typical use |
|---|---|---|---|
| SSS | Three sides (a, b, c) | Law of Cosines for angles, Heron for area | Surveying, construction measurements, known side lengths |
| SAS | Two sides + included angle | Law of Cosines to find third side, then Law of Sines | Engineering drawings, problems with an included angle |
| ASA/AAS | Two angles + one side | Angle sum then Law of Sines | Compass/heading geometry, many textbook trig problems |
SSS: Three Sides Known
If you know all three sides, you can find each angle using the Law of Cosines. For example, angle A depends on sides b and c and the opposite side a:
Once the angles are computed, the calculator finds area using Heron’s formula, which is convenient because it uses only side lengths. This is a common workflow for SSS problems.
SAS: Two Sides and the Included Angle
In SAS, you know two sides that “meet” at a known angle. The first step is usually finding the third side with the Law of Cosines:
Once the missing side is known, the triangle is effectively reduced to an SSS triangle, and the remaining angles can be computed (often using the Law of Sines for convenience).
ASA/AAS: Two Angles and One Side
With two angles, the third angle is determined immediately:
After that, use the Law of Sines to compute the unknown sides. The calculator’s ASA/AAS mode asks for angles A and B and a known side a (opposite A), then derives b and c from the ratio a/sin(A).
Right Triangle Calculations (Pythagorean Theorem + Trig)
Right triangles are a special (and extremely common) triangle type with one angle equal to 90°. They are used for ramps, stairs, roof pitch, diagonals, and distance calculations. When a triangle is right, the relationships simplify dramatically:
The right triangle tab lets you enter any two independent values (for example: both legs, or one leg and hypotenuse, or one leg and an acute angle) and compute the rest. It also provides a useful derived value: the altitude to the hypotenuse (often used in geometry proofs and similar triangles).
Area and Perimeter: The Three Most Practical Area Formulas
People often search for a triangle calculator because they need area quickly. This tool includes three widely used methods:
- Base × height: best when you know a perpendicular height (A = ½bh).
- Heron’s formula: best when you know all three sides.
- Two sides + included angle: best when you know an included angle (A = ½ab·sin(C)).
A common mistake is to use a “height” that isn’t perpendicular to the chosen base. If your height is not measured at 90° to the base, the base-height formula will be wrong. In those situations, Heron’s formula or the sine-area formula is usually safer.
Derived Measures: Inradius, Circumradius, Altitudes, and Medians
Once a triangle is solved (you know a, b, c and A, B, C), you can compute many helpful properties:
- Perimeter: P = a + b + c
- Semiperimeter: s = (a + b + c)/2
- Altitudes: ha = 2·Area/a, hb = 2·Area/b, hc = 2·Area/c
- Inradius: r = Area / s
- Circumradius: R = abc / (4·Area)
- Medians: ma = ½√(2b² + 2c² − a²), similarly for mb, mc
These values matter in design (fit circles inside/outside triangles), geometry problems, and quality checks (for example, whether a triangle is close to equilateral or highly skewed).
Triangle Classification (Side-based and Angle-based)
This calculator reports two common classifications:
- By sides: equilateral (all equal), isosceles (two equal), scalene (all different).
- By angles: acute (all < 90°), right (one = 90°), obtuse (one > 90°).
Angle classification can be determined from the computed angles, or in SSS mode by checking the Law of Cosines relationship. In practice, your rounding settings can slightly affect a “near-right” triangle, so it’s smart to round consistently with your assignment or engineering tolerance.
Common Mistakes to Avoid
- Invalid side sets: forgetting triangle inequality (a + b must be greater than c).
- Angles not summing to 180°: ASA/AAS inputs must leave room for the third angle.
- Wrong angle units: degrees vs radians (this tool uses degrees).
- Using non-included angle for SAS: SAS requires the included angle between the two known sides.
- Rounding too early: keep precision during intermediate steps, then round at the end.
FAQ
Triangle Calculator – FAQs
Answers about solving triangles, choosing formulas, and interpreting results.
Solving a triangle means finding all missing side lengths and angles, and often also derived values like perimeter, area, altitudes, and radii (inradius and circumradius).
Common solvable sets include SSS (three sides), SAS (two sides and included angle), and ASA/AAS (two angles and one side). For right triangles, any two independent values are usually enough.
For sides a, b, c, a valid triangle must satisfy a + b > c, a + c > b, and b + c > a. If any fails, no triangle exists with those sides.
Both use two angles and one side. In ASA the known side is between the two known angles; in AAS the known side is not between them. Either way, you find the third angle and then use the Law of Sines.
Common formulas include A = (1/2)·base·height, Heron’s formula A = √(s(s−a)(s−b)(s−c)), and A = (1/2)ab·sin(C) for two sides with included angle.
No. A triangle’s angles sum to 180°, so having two angles ≥ 90° would exceed 180°.
SSA (two sides and a non-included angle) can sometimes produce two different triangles (or none). This calculator focuses on SSS, SAS, and ASA/AAS for unambiguous solving.
The circumradius R is the radius of the circle through all three vertices; the inradius r is the radius of the inscribed circle touching all sides. They can be derived from sides/area.
This calculator uses degrees (°) for input and output.
Results are computed in the browser using standard floating-point arithmetic. Use the decimals setting to match your rounding requirements.