BrainyCalculators
Login
Advertisement

🔺 Triangle Calculator

Solve triangles with SSS, SAS, ASA, and right-triangle rules: missing sides, angles, area, and perimeter.

Triangle Sides, Angles, Area — SSS, SAS, ASA

BrainyCalculators editorial insight — unique to this tool

Law of cosines solves scalene when three sides known; Heron's formula area from three sides without height. Indian land plots and roof trusses use triangle geometry daily. Sum of angles always 180° in Euclidean plane.

When to use this calculator

Use for triangle-specific geometry. General 3D volumes → Volume calculator.

Evaluating sin, cos, or tan on an angle?

This page solves full triangles. For trig function evaluation, use the Trigonometry Calculator →

Area = ½ × base × height. Enter the base and perpendicular height.

What is a Triangle Calculator?

A triangle calculator solves for unknown sides and angles given partial information using law of sines, cosines, and Pythagoras where applicable.

Use this page when triangle side/angle data is the problem. Trigonometry evaluates sin/cos/tan on a single angle without full triangle solving.

Perimeter sums known sides; this page finds missing sides first when needed.

Triangle Formulas

Base & Height: Area = ½ × b × h
Heron's (SSS): s = (a+b+c)/2  |  Area = √(s(s−a)(s−b)(s−c))
SAS: Area = ½ × a × b × sin(C)
Law of Cosines: cos A = (b²+c²−a²) / (2bc)

Worked Example — Sides 3, 4, 5 (SSS)

s = (3 + 4 + 5) / 2 = 6
Area = √(6 × 3 × 2 × 1) = √36 = 6
Perimeter = 3 + 4 + 5 = 12
Angle A = arccos((16+25−9)/(2×4×5)) = arccos(0.8) ≈ 36.87°
Angle B = arccos((9+25−16)/(2×3×5)) = arccos(0.6) ≈ 53.13°
Angle C = 90° — Right Triangle

How the Triangle Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

  1. Read the values and operation selected.
  2. Normalize signs, decimals, fractions, or units if needed.
  3. Apply the mathematical rule or formula.
  4. Format the answer and any intermediate values for checking.

Assumptions and Limits

  • Inputs must be within the supported domain of the operation.
  • Decimal answers may be rounded for readability.
  • Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

The most common formula is Area = ½ × base × height, where height is the perpendicular distance from the base to the opposite vertex. When only sides are known, use Heron's formula.

Heron's formula computes a triangle's area using only its three side lengths. First calculate the semi-perimeter: s = (a+b+c)/2. Then Area = √(s(s−a)(s−b)(s−c)). It works for any triangle.

Use the law of cosines when you have SSS or SAS — it directly finds the missing side or angle. Use the law of sines (a/sin A = b/sin B) when you have AAS or ASA, as it is algebraically simpler in those cases.

By sides: Equilateral (all three sides equal, all angles 60°), Isosceles (two sides equal), Scalene (all sides different). By angles: Right (one 90° angle), Acute (all angles < 90°), Obtuse (one angle > 90°).

Real-World Applications

📐
Surveying & Land Measurement
Land surveyors use triangulation to measure large distances and areas without direct measurement — placing survey stations at known positions and measuring angles to a target point using a theodolite, then applying the Law of Sines to calculate the distance to the target. The triangle calculator supports these field calculations by solving for unknown sides given measured angles and one baseline distance, the fundamental operation of triangulation-based surveying.
🏗️
Roof Truss & Structural Design
Structural engineers design roof trusses, bridge frameworks, and space frames by resolving them into networks of triangular elements — triangles being the only polygon that is inherently rigid under load. The triangle calculator determines the exact lengths of structural members (rafters, ties, diagonals) given the span, pitch angle, and rise of a roof, enabling material quantity calculations and load distribution analysis in simple residential and commercial structures.
🗺️
Navigation & Bearing Calculation
Sailors using celestial navigation, hikers using triangulation to fix position from two known bearings, and pilots using radio navigation waypoints all apply triangle mathematics. Given two known reference points and the bearing angle from a current position to each, the triangle formed by the observer and two reference points can be solved using the Law of Sines to determine the observer's position. The triangle calculator performs this computation from angle and baseline inputs.
🎮
3D Computer Graphics & Game Development
All surfaces in real-time 3D graphics (games, virtual reality, architectural visualisation) are represented as triangular meshes — collections of triangles that approximate curved surfaces. The rendering engine calculates triangle area, normal vectors, and intersection tests for every triangle in the scene at frame rates of 60–120 Hz. Triangle calculators and the underlying mathematics are at the core of GPU architecture, 3D modelling tools, and game physics engines.
🔭
Astronomy & Stellar Distance Measurement
Astronomers use stellar parallax — the apparent shift in a star's position against background stars as Earth moves through its orbit — to measure distances to nearby stars. The parallax angle and the Earth-Sun baseline form a right triangle; the triangle calculator converts the measured parallax angle (in arcseconds) to the star's distance in parsecs using the relationship: distance (pc) = 1 / parallax angle (arcseconds). This is the foundation of the cosmic distance ladder.
⚡
Electrical Engineering — Impedance Triangles
AC electrical circuits containing resistance, inductance, and capacitance are analysed using impedance triangles — right triangles where the hypotenuse represents total impedance (Z), and the two legs represent resistance (R, real) and reactance (X, imaginary). The triangle calculator computes the impedance magnitude and phase angle from R and X values, or decomposes a known impedance into its resistive and reactive components — operations fundamental to filter design and power factor correction calculations.

Common Mistakes

1
Applying the Pythagorean theorem to non-right triangles
The Pythagorean theorem (c² = a² + b²) applies only to right-angled triangles — triangles with exactly one 90° angle. Applying it to acute or obtuse triangles produces incorrect results. The generalised version for any triangle is the Law of Cosines: c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when C = 90° (cos 90° = 0). The first diagnostic question when using triangle formulas should always be: "Is this a right triangle?" — only then is the Pythagorean theorem applicable.
2
The ambiguous case in the Law of Sines (SSA input)
When solving a triangle from two sides and a non-included angle (SSA), the Law of Sines may produce two valid triangles, one valid triangle, or no valid triangle — the "ambiguous case." Given sides a and b and angle A: if a ≥ b, there is one solution; if a < b, there may be two solutions (when a < b and a > b·sin(A)), one solution (when a = b·sin(A)), or no solution (when a < b·sin(A)). The triangle calculator handles this by checking both possible solutions and presenting them where both are geometrically valid.
3
Forgetting that angle sum = 180° can verify any triangle solution
After solving a triangle, the three angles must sum to exactly 180°. This is an immediate sanity check — if the three computed angles sum to more or less than 180° (within floating-point precision), the calculation contains an error. Many errors in manual triangle solving (using the wrong formula, applying a law of sines incorrectly, introducing a unit error) are caught immediately by this simple verification step.
4
Confusing the included angle (SAS) with a non-included angle (SSA)
In SAS configuration, the known angle is between the two known sides — the angle is sandwiched between sides a and b, so C is the angle at vertex C (between sides a and b). If the known angle is not the included angle but is opposite one of the known sides, the configuration is SSA (the ambiguous case), which requires the Law of Sines rather than the Law of Cosines. Misidentifying an SSA as SAS applies the Law of Cosines where the Law of Sines is needed, producing an incorrect result.
5
Mixing degrees and radians in trigonometric calculations
Scientific calculators and programming languages need to be explicitly told whether angles are in degrees or radians. The Law of Cosines formula c² = a² + b² − 2ab·cos(C) requires cos(C) to be evaluated with C in the correct unit. Most programming environments default to radians: if the angle is 45° but is supplied without converting to radians (π/4 ≈ 0.785), cos(45) in radians computes a different result than cos(45°). Always verify which angle mode is active before computing trigonometric functions.

Triangle Solving Method by Input Combination

Known Method Notes
SSS (3 sides) Law of Cosines Fully determined, 1 solution
SAS (2 sides + included angle) Law of Cosines Fully determined, 1 solution
ASA / AAS (2 angles + 1 side) Law of Sines Fully determined, 1 solution
SSA (2 sides + non-included angle) Law of Sines Ambiguous: 0, 1, or 2 solutions
AAA (3 angles only) N/A Infinite solutions — scale unknown

References

  1. Euclid. Elements, Book I. c. 300 BCE.
  2. Stewart, J. Trigonometry. Cengage Learning, 2016.
  3. Larson, R. Precalculus: Real Mathematics, Real People. Cengage, 2019.
  4. Weisstein, E.W. Law of Cosines. MathWorld — Wolfram Research, 2024.
  5. NCTM. Principles to Actions: Ensuring Mathematical Success for All. nctm.org, 2014.

Related Calculators

Browse all Math calculators →
📐 Math

Pythagorean Theorem Calculator

Calculate the missing side of a right triangle using the Pythagorean theorem.

Calculate now
📐 Math

Trigonometry Calculator

Calculate sin, cos, tan, and their inverses in degrees or radians.

Calculate now
⬛ Math

Area Calculator

Calculate the area of any shape — circle, triangle, rectangle, trapezoid, and more.

Calculate now