Triangle Solver — SSS, SAS, ASA/AAS & Right Triangle

Guide

About the Triangle Solver

Last updated: July 2026 · Reviewed by the NeftCal editorial team

This triangle solver works out every missing angle, side, area, and perimeter of a triangle no matter which three measurements you start with. Choose SSS if you know all three sides, SAS if you know two sides and the angle between them, ASA/AAS if you know two angles and a side, or Right Triangle if you know two legs or a leg and the hypotenuse. Behind the scenes it applies the Law of Cosines, Law of Sines, Heron's formula, and the Pythagorean theorem automatically, and also classifies the result — scalene, isosceles, or equilateral by side length, and acute, right, or obtuse by angle — so you get the complete picture in one place.

What This Calculator Measures

For every mode, the solver finds the three interior angles, the three side lengths, the perimeter (their sum), and the area — using Heron's formula for SSS, the ½ab·sin(C) formula for SAS and ASA/AAS, and ½ × base × height for right triangles. It then classifies the triangle by comparing side lengths to each other and the largest angle to 90°.

Who Should Use This Calculator

Geometry, trigonometry, and precalculus students who want a single tool covering every triangle case, construction and roofing professionals working out rafter lengths and pitch angles, land surveyors solving triangular parcels, DIYers and hobbyists building triangular frames or supports, and anyone who has three triangle measurements — of any kind — and wants the full solution without picking which formula applies.

Why a General Triangle Solver Matters

Real triangles rarely arrive labeled with "use the Law of Cosines here." Knowing which of SSS, SAS, ASA/AAS, or a right-triangle shortcut applies to your specific set of known values — and then executing the correct formula without error — is exactly the kind of repetitive, mistake-prone task a single unified solver removes, while also adding the area, perimeter, and classification most standalone formula calculators skip.

Real-World Applications

Contractors compute rafter length and roof pitch angles from known rise, run, and slope measurements. Land surveyors solve triangular parcel boundaries from measured sides and angles. Engineers verify truss and gusset geometry across all four input cases in one tool. Students check geometry and trigonometry homework covering every triangle type. Hobbyists and makers design triangular braces, shelving brackets, and frames with confidence in the final dimensions.

Tips for Accurate Results

Pick the tab that matches what you actually measured — mixing up SAS's "included" angle with a non-included one gives an incorrect result with no warning.
In Right Triangle mode, double-check whether you're entering two legs or a leg and the hypotenuse — the hypotenuse is always the longest side, opposite the right angle.
Use the classification result (e.g. "isosceles, right") as a quick sanity check — if it doesn't match your intuition about the shape, re-check your inputs.

Formula

The Triangle Solver Formulas, Explained

Every formula behind the four solving modes

SSS — Law of Cosines + Heron's Formula

A = acos((b²+c²−a²)/(2bc)) | B = acos((a²+c²−b²)/(2ac)) | C = 180° − A − B
s = (a+b+c)/2 | Area = √(s(s−a)(s−b)(s−c))

SAS — Law of Cosines

c = √(a²+b²−2ab·cos(C)) | A = asin(a·sin(C)/c) | B = 180° − C − A | Area = ½ab·sin(C)

ASA / AAS — Law of Sines

C = 180° − A − B | b = a·sin(B)/sin(A) | c = a·sin(C)/sin(A) | Area = ½ab·sin(C)

Right Triangle — Pythagorean Theorem

Two legs known: c = √(a²+b²) | Leg + hypotenuse known: b = √(c²−a²) | Area = ½ab

Every Mode

Perimeter = a + b + c | Sides classification: equilateral / isosceles / scalene | Angle classification: acute / right / obtuse (based on the largest angle vs 90°)

🔺

Four Cases, One Tool

SSS, SAS, ASA/AAS, and right-triangle inputs each route to the correct formula automatically.

📐

Area & Perimeter Every Time

Regardless of mode, the solver always reports area (via the matching formula) and perimeter (a+b+c).

🏷️

Automatic Classification

Every result is labeled by side type (scalene/isosceles/equilateral) and angle type (acute/right/obtuse).

⚙️ Why This Formula Works

Each mode uses the minimum formula needed for that specific combination of known values: the Law of Cosines when no matching angle-side pair exists (SSS, SAS), the Law of Sines when one does (ASA/AAS), and the Pythagorean theorem for the special case of a 90° angle (Right Triangle). Heron's formula is a purely algebraic rearrangement of the Law of Cosines-based area formula that needs no angle at all.

🎯 When to Use Each Mode

SSS: you measured all three sides directly
SAS: you know two sides and the angle between them
ASA/AAS: you know two angles and one side
Right Triangle: you know it's a right triangle and have two legs, or a leg and the hypotenuse

📋 Assumptions

All side lengths are positive real numbers, and all angles are given in degrees
The triangle lies in flat (Euclidean) geometry, where angles sum to 180°
In Right Triangle mode, the given right angle is exactly 90°

⚠️ Limitations of the Formula

SSS and Right Triangle (leg+hypotenuse) inputs are rejected if they violate the triangle inequality or Pythagorean constraint
This tool does not cover the ambiguous SSA case — see our dedicated Law of Sines Calculator for that
Very thin (near-degenerate) triangles can amplify floating-point rounding error in the angle results

Walkthrough

Step-by-Step: How to Use the Triangle Solver

From choosing a mode to reading the full solved triangle

Choose your case

Select SSS, SAS, ASA/AAS, or Right Triangle based on which three measurements you already know.

Enter your known values

Type in the sides and/or angles for your chosen mode. For Right Triangle mode, pick "Two Legs" or "Leg + Hypotenuse" first.

Click "Calculate"

The solver validates your inputs, then applies the correct formula (Law of Cosines, Law of Sines, Heron's formula, or the Pythagorean theorem).

Read every angle and side

Review all three angles and all three sides of the fully solved triangle.

Check area, perimeter & classification

See the computed area, perimeter, and the triangle's classification (e.g. "scalene, acute" or "isosceles, right").

Example

Worked Example: SSS Mode

Solving a triangle with sides 5, 6, and 7 — angles, area, perimeter, and classification

Scenario

A triangular garden bed has three measured sides: a = 5, b = 6, c = 7 meters. What are its angles, area, perimeter, and classification?

Side a5 m

Side b6 m

Side c7 m

Step 1 — Check the triangle inequality: 5+6=11>7, 5+7=12>6, 6+7=13>5 — valid triangle.

Step 2 — Solve the angles: A = acos((36+49−25)/84) ≈ 44.4153°. B = acos((25+49−36)/70) ≈ 57.1217°. C = 180−44.4153−57.1217 ≈ 78.4630°.

Step 3 — Solve the area with Heron's formula: s = (5+6+7)/2 = 9. Area = √(9×(9−5)×(9−6)×(9−7)) = √(9×4×3×2) = √216 ≈ 14.6969 m².

Step 4 — Solve the perimeter: P = 5+6+7 = 18 m.

Step 5 — Classify: all three sides differ → scalene; the largest angle (78.4630°) is less than 90° → acute.

Angle A

44.4153°

Angle B

57.1217°

Angle C

78.4630°

Area

14.6969 m²

Perimeter

18 m

Classification

Scalene, Acute

Explanation: Starting from three sides alone (SSS) was enough to recover every angle, the area via Heron's formula, the perimeter, and a complete classification — no angle measurement was needed anywhere in the process.

Interpretation

Understanding Your Triangle Solver Results

What each output means across all four modes

Output	What It Represents	Notes
Angle A, B, C	The three interior angles	Always sum to exactly 180°
Side a, b, c	The three side lengths	Each must be positive; largest side is opposite the largest angle
Area	The two-dimensional space enclosed by the triangle	Always in squared units (e.g. m²)
Perimeter	The total distance around the triangle	Equal to a + b + c, in linear units
Side classification	Scalene, isosceles, or equilateral	Based purely on comparing side lengths
Angle classification	Acute, right, or obtuse	Based on comparing the largest angle to 90°

Reading the results together: the two classifications (side-based and angle-based) are independent — a triangle can be, for example, isosceles and right, or scalene and obtuse, in any combination except that an equilateral triangle is always acute.

Typical ranges: all sides must be positive, all angles must fall strictly between 0° and 180° and sum to 180°, and area and perimeter must both be positive.

Manual verification: once you have all three sides and angles, recompute the area a second way (e.g. Heron's formula from the sides, or ½ab·sin(C) from two sides and an angle) — both should agree closely, confirming the solved triangle is internally consistent.

Use Cases

Practical Use Cases for the Triangle Solver

Where a single tool covering every triangle case is genuinely useful

🎓

Geometry & trigonometry homework

Check SSS, SAS, ASA/AAS, and right-triangle problems in one place.

🏠

Roof pitch & rafter length

Solve the rise/run/rafter right triangle for accurate roofing cuts.

🗺️

Land surveying

Solve triangular parcel boundaries from any combination of sides and angles.

🏗️

Construction & framing

Verify triangular brace, gusset, or truss dimensions before cutting material.

🪜

Ladder & ramp angle safety

Check the right-triangle geometry of a ladder's height, base distance, and length.

🛠️

DIY & woodworking

Design triangular shelf brackets, braces, and frames with confidence in the cut dimensions.

🏛️

Architecture & drafting

Check triangular facade, gable, or truss elements against design specifications.

⛵

Navigation & triangulation

Solve course triangles from any mix of known legs, angles, or bearings.

📊

Physics & vector problems

Resolve force, velocity, or displacement triangles regardless of which values are known.

🖥️

Computer graphics & game dev

Solve triangle geometry for meshes, collision shapes, and procedural generation.

📐

Area & material estimation

Estimate paint, flooring, or land coverage from a triangular area calculation.

🧭

Robotics & sensor geometry

Solve link-and-joint triangles for arm reach and positioning calculations.

🧪

Classifying unknown triangles

Quickly determine whether a triangle is scalene/isosceles/equilateral and acute/right/obtuse.

Pros & Cons

Advantages and Limitations

What this triangle solver does well, and where it doesn't apply

✅ Advantages

Free, instant, and requires no signup or account
Runs entirely in your browser — no data ever leaves your device
Covers four input cases — SSS, SAS, ASA/AAS, and Right Triangle — in a single tool
Always reports area, perimeter, and a full side/angle classification
Automatically checks the triangle inequality and Pythagorean validity
Removes the need to remember which formula applies to which case
Useful across construction, surveying, engineering, and classroom contexts alike
Accepts decimal inputs for precise, real-world measurements
Clearly separates each mode with dedicated tabs and labeled inputs
Fast-loading and fully mobile-friendly
Consistent, error-free results every time
Free to use as many times as needed, with no calculation limit

⚠️ Limitations

Does not cover the ambiguous SSA case — see our dedicated Law of Sines Calculator for that
SSS and leg+hypotenuse inputs that fail their geometric constraints are rejected with no partial result
Assumes flat (Euclidean) geometry — not valid for spherical or curved-surface triangles
Right Triangle mode assumes the entered angle is exactly 90° — it does not verify this independently
Displays decimal results rounded to four places, which can hide tiny precision loss
Does not draw or visually plot the resulting triangle
Cannot detect a mismatched side/angle pairing (e.g. entering a non-included angle in SAS mode)

Reference

Comparing the Four Solving Methods

SSS, SAS, ASA/AAS, and Right Triangle side by side

Mode	Known Values	Core Formula	Area Formula
SSS	All three sides	Law of Cosines (for angles)	Heron's formula
SAS	Two sides + included angle	Law of Cosines (for third side)	½ab·sin(C)
ASA / AAS	Two angles + one side	Law of Sines	½ab·sin(C) after solving sides
Right Triangle	Two legs, or leg + hypotenuse	Pythagorean theorem	½ × leg × leg

Common Mistakes and Expert Tips

❌ Common Mistakes

Choosing SAS mode but entering an angle that isn't actually included between the two known sides
Confusing the ambiguous SSA case with ASA/AAS — they require very different solving logic
Forgetting to check the triangle inequality before assuming three sides can form a triangle
Entering the hypotenuse as one of the "legs" in Right Triangle mode
Mixing degrees and radians when entering angle values
Assuming area formulas are interchangeable across modes without matching the known values correctly

💡 Expert Tips & Best Practices

If your case is specifically SSA (two sides + a non-included angle), use the Law of Sines Calculator instead — it correctly handles the ambiguous case
For SAS or SSS specifically, the dedicated Law of Cosines Calculator offers the same math with a narrower, focused interface
Use the classification result as a fast sanity check against your mental picture of the triangle's shape
In Right Triangle mode, remember the hypotenuse is always the longest side and is always opposite the 90° angle
Cross-check the perimeter and area against a rough sketch or physical measurement before finalizing a design

📝

Summary: This triangle solver handles SSS, SAS, ASA/AAS, and right-triangle inputs in one place, always returning area, perimeter, and classification alongside every angle and side. For the ambiguous SSA case specifically, use the Law of Sines Calculator; for a narrower SAS/SSS-only tool, see the Law of Cosines Calculator.

FAQ

Frequently Asked Questions

Common questions about solving triangles

What triangle cases can this solver handle?

Four cases: SSS (all three sides), SAS (two sides and the included angle), ASA/AAS (two angles and a side), and Right Triangle (two legs, or one leg and the hypotenuse). Together these cover every standard way a triangle can be specified with three known measurements.

How does the solver compute area for each mode?

For SSS, it uses Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter. For SAS and ASA/AAS, once two sides and the angle between them are known, Area = ½ab·sin(C). For a right triangle, Area is simply half the product of the two legs.

What is Heron's formula?

Heron's formula finds a triangle's area from its three side lengths alone, with no angle needed: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. For sides 5, 6, 7, s = 9 and Area = √(9×4×3×2) = √216 ≈ 14.6969.

How does the solver classify a triangle by its sides?

If all three sides are equal, it's equilateral. If exactly two sides are equal, it's isosceles. If all three sides are different lengths, it's scalene. This classification is purely about side lengths and is independent of the angle-based classification.

How does the solver classify a triangle by its angles?

It compares the largest interior angle to 90°. If the largest angle equals 90°, the triangle is a right triangle. If it's greater than 90°, the triangle is obtuse. If every angle is less than 90°, the triangle is acute.

What is the difference between SSS, SAS, and ASA/AAS?

SSS means all three sides are known and no angle. SAS means two sides plus the angle directly between them are known. ASA/AAS means two angles plus one side are known (the side can be between the angles or opposite one of them — both use the same solving method).

How does the Right Triangle mode work?

You can enter either two legs (the two sides forming the right angle) or one leg plus the hypotenuse. Given two legs a and b, the hypotenuse is c = √(a² + b²) via the Pythagorean theorem. Given one leg a and the hypotenuse c, the other leg is b = √(c² − a²).

Why does my SSS or right-triangle input get rejected?

SSS inputs are checked against the triangle inequality — each side must be shorter than the sum of the other two, or no triangle exists. In right-triangle mode with a leg and hypotenuse, the hypotenuse must be strictly longer than the given leg, or the remaining leg would not be a real number.

What is the perimeter and how is it calculated?

The perimeter is simply the sum of all three sides: P = a + b + c. It's computed the same way regardless of which mode (SSS, SAS, ASA/AAS, or Right Triangle) you use, once all three sides are known.

Can a triangle be both isosceles and right?

Yes. A triangle with two equal legs and a 90° angle between them — for example, legs of 5 and 5 with a hypotenuse of 5√2 ≈ 7.071 — is both isosceles (two equal sides) and right (one 90° angle). The side-based and angle-based classifications are independent of each other.

Can an equilateral triangle be solved with this tool?

Yes — enter all three equal sides in SSS mode, and the solver will confirm all three angles are exactly 60° and classify the triangle as equilateral and acute. An equilateral triangle can never be a right or obtuse triangle.

Which mode should I use if I only know two sides and one angle?

It depends on which angle you know. If the angle is directly between the two known sides, use SAS. If the angle is opposite one of the two sides instead, that's actually the SSA case handled on our dedicated Law of Sines Calculator, since it can be ambiguous (zero, one, or two solutions).

Does this solver work for obtuse triangles?

Yes, all four modes correctly solve obtuse triangles. The Law of Cosines-based methods (SSS, SAS) handle obtuse angles natively, and the classification logic will correctly label the triangle as obtuse whenever its largest angle exceeds 90°.

How accurate are the results from this calculator?

Results are computed with full floating-point precision using standard trigonometric functions and Heron's formula, then displayed rounded to four decimal places. For most practical uses — construction, homework, or design — this level of precision is more than sufficient.

Learn More

Authoritative Resources on Solving Triangles

Trusted educational references to go deeper on triangle geometry

🎓 Khan Academy — TrigonometryFree lessons and practice solving triangles ↗ 📖 OpenStax Precalculus 2eFree, peer-reviewed college textbook covering all triangle cases ↗ 🔬 Wolfram MathWorld — Trigonometric FunctionsFormal mathematical definitions and identities ↗

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