Every bridge you have ever crossed is, at heart, a collection of triangles. So is the roof over your head, the crane on the horizon, the bracket holding up your bookshelf, and the invisible web of measurements a surveyor throws across a valley. The triangle is the only polygon that cannot be pushed out of shape without bending a side, which is why engineers trust it, why ancient builders squared their foundations with knotted ropes, and why every trigonometry course on Earth begins with it. It is also why, sooner or later, almost everyone ends up needing to solve one.
The triangle calculator on this page does exactly that, completely free and entirely in your browser. Give it any valid combination of three values — three sides, two sides and an angle, or two angles and a side — and it returns everything there is to know about the triangle: all three sides, all three angles, the area, the perimeter, the three heights, the medians, the angle bisectors, the inradius and circumradius, the triangle’s classification, and a diagram drawn to true scale so you can see the shape you just described with numbers. It speaks six languages, works in degrees, radians or gradians, supports eight length units, and never sends a single byte of your data anywhere.
This article is the long-form companion to the tool. It walks through the geometry a triangle solver depends on, the exact formulas running under the hood (including two numerical tricks most calculators skip), quick-reference conversion tables you can bookmark, the special triangles worth memorizing, four fully worked examples you can reproduce in seconds, and answers to the questions people actually type into Google at eleven o’clock the night before a test. Whether you are a student staring at an SSA problem, a carpenter checking a corner, or a webmaster who wants this calculator on your own site, everything you need is below.
What Makes This Triangle Calculator Different
There is no shortage of triangle solvers online, so it is fair to ask why this one deserves a spot in your bookmarks. The short answer is that we built it the way an engineer would want it built, then tested it the way a paranoid engineer would test it — more than a hundred automated checks run against known-exact triangles before it ever reached this page.
The longer answer comes down to five decisions. First, it handles all five input methods — SSS, SAS, ASA, AAS and SSA — including the notorious ambiguous case, where it will honestly show you both valid triangles side by side instead of quietly picking one. Second, it uses numerically stable mathematics: Heron’s formula in the rearranged form recommended by William Kahan, and angles computed through the half-angle identity rather than a raw inverse cosine, so results stay accurate even for extremely flat, needle-thin triangles that make ordinary calculators wobble. Third, it draws every solution to scale on a graph-paper grid, labels the vertices, sides and angles, and automatically marks a right angle with the little square symbol your geometry teacher used. Fourth, it is genuinely international: six interface languages with full right-to-left support for Arabic, plus input that accepts comma decimals and even Eastern Arabic and Devanagari digits. And fifth, everything runs locally on your device — no account, no ads, no data collection, no waiting on a server.
Triangle Basics: Sides, Angles and Notation
A triangle is three points — the vertices — joined by three straight segments, the sides. That is the whole definition, and yet it generates an entire branch of mathematics. To talk about a specific triangle without drowning in words, mathematicians use a naming convention this calculator follows exactly: the vertices are labeled with capital letters A, B and C, and each side takes the lowercase letter of the vertex it faces. Side a sits opposite angle A, side b opposite angle B, side c opposite angle C. Once that pairing clicks, every formula in trigonometry becomes readable at a glance, and you will always know which box on the calculator wants which number.
One consequence of the pairing is worth engraving somewhere: the largest angle always faces the longest side, and the smallest angle faces the shortest side. A triangle cannot hide its proportions. If someone tells you the sides are 2, 6 and 7, you already know, before computing anything, that the angle opposite the 7 is the big one.
The 180° Rule and the Triangle Inequality
Two laws govern whether a triangle can exist at all. The first is the angle sum: in flat (Euclidean) geometry, the three interior angles of every triangle add up to exactly 180° — equivalently π radians, or 200 gradians. Tear the three corners off a paper triangle and line them up along a ruler’s edge; they form a perfect straight line every time. The calculator leans on this rule constantly: whenever it knows two angles, the third is simply 180° minus their sum, which is also why the angles it reports always total 180° to the last digit rather than 179.9998°.
The second law is the triangle inequality: the sum of any two sides must be strictly greater than the third side. Sides of 3, 4 and 9 cannot close into a triangle, because 3 + 4 = 7 falls short of 9 — the two short sticks simply cannot reach each other across the long one. Enter such values in SSS mode and the tool will tell you plainly that the sides do not form a valid triangle, instead of inventing an answer. A surprising number of “my calculator is broken” moments are really the triangle inequality doing its job.
The Six Types of Triangles
Every triangle carries two labels at once: one describing its sides, one describing its angles. The calculator reports both automatically — a result like Isosceles • Acute or Scalene • Right — using a small numerical tolerance so that a triangle you clearly meant to be equilateral is recognized as such even after rounding.
| Classified by | Type | Definition | Quick fact |
|---|---|---|---|
| Sides | Equilateral | All three sides equal | All angles are exactly 60° |
| Sides | Isosceles | Exactly two sides equal | The two base angles are equal |
| Sides | Scalene | All three sides different | All three angles are different too |
| Angles | Acute | All angles less than 90° | The circumcenter lies inside the triangle |
| Angles | Right | One angle exactly 90° | The Pythagorean theorem applies: a² + b² = c² |
| Angles | Obtuse | One angle greater than 90° | The circumcenter lies outside the triangle |
Five Ways to Solve a Triangle: SSS, SAS, ASA, AAS and SSA
“Solving” a triangle means starting from a few known measurements and finding all the rest. Geometry proves that three well-chosen pieces of information pin a triangle down completely — these are the classic congruence criteria you may remember from school — and each combination gets its own tab in the calculator. Three angles alone (AAA) are famously not enough: they fix the shape but not the size, which is why every similar triangle from a postage stamp to a football field shares the same three angles. That is the one combination no honest solver can offer.
| Mode | You provide | Main engine | Possible solutions |
|---|---|---|---|
| SSS | All three sides (a, b, c) | Law of cosines / half-angle identity | One, if the triangle inequality holds |
| SAS | Two sides and the angle between them (b, A, c) | Law of cosines | Always exactly one |
| ASA | Two angles and the side between them (B, a, C) | Angle sum + law of sines | One, if the two angles total less than 180° |
| AAS | Two angles and a non-included side (A, B, a) | Angle sum + law of sines | One, if the two angles total less than 180° |
| SSA | Two sides and a non-included angle (a, b, A) | Law of sines, checked both ways | Zero, one, or two — the ambiguous case |
SSS — Three Sides
This is the mode for anyone holding a tape measure. You know how long the three edges are — a garden bed, a gable, a sail — and you want the angles and the area. The calculator first verifies the triangle inequality, then recovers each angle from the side lengths and computes the area with Heron’s formula. It is the workhorse mode of carpentry and land measurement, and the one behind the famous 3-4-5 check we will meet again below.
SAS — Two Sides and the Included Angle
Picture two fence rails meeting at a post: you know each rail’s length and the angle of the corner, and you want the straight-line distance between the far ends. That is SAS. The law of cosines produces the missing third side directly, after which the remaining angles follow. Because two sides and the angle between them lock a triangle rigidly, SAS always yields exactly one answer — no ambiguity, ever.
ASA — Two Angles and the Included Side
This is the surveyor’s mode. Measuring long distances is hard; measuring angles is easy, which is why the theodolite was invented. Stake out one baseline you can measure, sight a distant target from both ends, record the two angles, and ASA delivers every other distance in the triangle without you ever crossing the river, the ravine or the property line. Worked example 3 below does exactly this.
AAS — Two Angles and Any Side
A close cousin of ASA: you still know two angles, but the side you measured is not the one between them — it sits opposite one of the known angles. Since two angles instantly reveal the third (the 180° rule again), the law of sines scales the whole triangle off that single side. In practice ASA and AAS feel identical to use; the calculator keeps them separate so the labels always match your textbook or your field notes.
SSA — The Ambiguous Case
Here is the mode with a reputation. You know two sides and an angle that is not between them — and, depending on the numbers, there may be no triangle, exactly one, or two genuinely different triangles that fit. It is not a flaw in the mathematics; it is a fact about geometry. Imagine swinging side a like a door hinged at the end of side b: for certain lengths, the swinging side can touch the ground in two distinct places, producing one triangle that is tall and acute-looking and another that is low and obtuse.
Most calculators silently return only one of the two, which is how students lose marks and, occasionally, how engineers get surprised. This tool checks both candidates against the 180° rule and, when both survive, displays Solution 1 and Solution 2 as complete, separately drawn triangles with a clear note explaining why. For the record, here is the full logic, assuming you enter sides a and b with angle A opposite side a:
| Given angle A | Condition | Number of triangles |
|---|---|---|
| Acute (A < 90°) | b × sin A > a | None — side a is too short to reach |
| Acute (A < 90°) | b × sin A = a | Exactly one, with a right angle at B |
| Acute (A < 90°) | b × sin A < a < b | Two distinct triangles — the ambiguous case |
| Acute (A < 90°) | a ≥ b | Exactly one |
| Right or obtuse (A ≥ 90°) | a ≤ b | None |
| Right or obtuse (A ≥ 90°) | a > b | Exactly one |
The Math Under the Hood: Law of Sines, Law of Cosines and Heron’s Formula
You never need to touch a formula to use the calculator, but knowing what it runs makes the results more trustworthy — and makes you faster at spotting your own mistakes on paper. Three theorems do nearly all the work.
The Law of Cosines
The law of cosines is the Pythagorean theorem grown up. For any triangle,
c² = a² + b² − 2ab × cos C
where C is the angle between sides a and b. Set C to 90° and cos C vanishes, collapsing the formula into the familiar a² + b² = c² — which is a nice reminder that a right triangle is not a special theorem, just a special angle. A geometric version of this relationship appears as far back as Euclid’s Elements. The calculator uses the law of cosines in SAS mode to find the missing side, and its rearranged form to recover angles from three known sides.
The Law of Sines
The law of sines is a statement of proportion:
a / sin A = b / sin B = c / sin C = 2R
Every side, divided by the sine of its opposite angle, gives the same number — and that number, elegantly, equals the diameter of the circle passing through all three vertices (twice the circumradius R). It is the engine of ASA, AAS and SSA mode: know one full side-angle pair and any other single value, and the proportion unlocks the rest. It is also the source of the ambiguous case, because two different angles — one acute, one obtuse — can share the same sine.
Heron’s Formula — and the Numerically Stable Version This Tool Runs
Heron of Alexandria, writing in the first century, left us a small miracle: the area of a triangle from its three sides alone, no height required. With the semi-perimeter s = (a + b + c) / 2,
Area = √( s (s − a)(s − b)(s − c) )
The classic formula is perfect on paper and slightly treacherous inside a computer. For very flat, needle-shaped triangles — say sides of 100000, 100000 and 0.0001 — the subtractions inside the root cancel almost completely, and floating-point arithmetic can shed most of its accuracy in the process. William Kahan, the numerical analyst behind the IEEE floating-point standard used by essentially every computer on Earth, published a rearrangement that avoids the cancellation. Sort the sides so that a ≥ b ≥ c, then compute:
Area = ¼ √( (a + (b + c)) × (c − (a − b)) × (c + (a − b)) × (a + (b − c)) )
The parentheses are not decoration — they are the algorithm, and this calculator evaluates them exactly as written. It applies the same philosophy to angles: instead of the raw inverse cosine, which loses precision for angles near 0° or 180°, it uses the well-conditioned half-angle identity tan(A / 2) = r / (s − a), where r is the inradius, evaluated with the two-argument arctangent. You will never notice any of this on an everyday triangle — and that is precisely the point. On the pathological ones, it is the difference between a right answer and a plausible-looking wrong one.
Every Measurement the Calculator Returns
Solve any triangle and the tool reports far more than three sides and three angles. The main panel shows the area, the perimeter and the triangle’s type; one tap on More measurements opens the full set. Here is everything on offer, with the exact formula behind each value — the same formulas the calculator evaluates, listed so you can verify any result by hand:
| Measurement | Label in the tool | Formula |
|---|---|---|
| Perimeter | Perimeter | P = a + b + c |
| Semi-perimeter | s | s = (a + b + c) / 2 |
| Area | Area | K = √( s (s − a)(s − b)(s − c) ) — stable Kahan form |
| Height to side a | h(a) | h(a) = 2K / a — likewise h(b) = 2K / b and h(c) = 2K / c |
| Median to side a | m(a) | m(a) = ½ √( 2b² + 2c² − a² ) — likewise for m(b), m(c) |
| Angle bisector from A | t(a) | t(a) = 2 √( b c s (s − a) ) / (b + c) — likewise for t(b), t(c) |
| Inradius (inscribed circle) | r | r = K / s |
| Circumradius (circumscribed circle) | R | R = a b c / (4K), equivalently R = a / (2 sin A) |
| Classification | Triangle type | By sides (equilateral / isosceles / scalene) and by angles (acute / right / obtuse) |
Two identities tie the table together beautifully and make excellent sanity checks: the area always equals r × s (inradius times semi-perimeter), and it always equals a b c / 4R. If you ever suspect a hand calculation, test it against both.
Quick Conversion Tables Built Into the Tool
The calculator lets you work in three angle units and eight length units, and it converts angle values live: type 30 in degree mode, switch the selector to radians, and the box itself updates to 0.5235988 — your number, translated, not erased. The tables below are the same conversions written out for reference, so this page works even when you are away from the tool with nothing but a pencil.
Degrees, Radians and Gradians
Degrees split a full turn into 360 parts — a convention usually traced to Babylonian base-60 astronomy. Radians are the mathematician’s unit: one radian is the angle whose arc equals the circle’s radius, which makes a full turn 2π and turns calculus formulas clean. Gradians (also written gon) split a right angle into exactly 100 parts; they were born of French metrication and survive today chiefly in surveying instruments, where decimal right angles are a genuine convenience. The three interconvert with fixed factors:
| Conversion | Multiply by | Numeric factor |
|---|---|---|
| Degrees → Radians | π / 180 | ≈ 0.0174533 |
| Radians → Degrees | 180 / π | ≈ 57.2957795 |
| Degrees → Gradians | 10 / 9 | ≈ 1.1111111 |
| Gradians → Degrees | 9 / 10 | 0.9 exactly |
| Radians → Gradians | 200 / π | ≈ 63.6619772 |
| Gradians → Radians | π / 200 | ≈ 0.0157080 |
And the angles you meet most often, in all three units at once:
| Degrees | Radians (exact) | Radians (decimal) | Gradians |
|---|---|---|---|
| 30° | π / 6 | 0.5236 | 33.333 gon |
| 45° | π / 4 | 0.7854 | 50 gon |
| 60° | π / 3 | 1.0472 | 66.667 gon |
| 90° | π / 2 | 1.5708 | 100 gon |
| 120° | 2π / 3 | 2.0944 | 133.333 gon |
| 135° | 3π / 4 | 2.3562 | 150 gon |
| 150° | 5π / 6 | 2.6180 | 166.667 gon |
| 180° | π | 3.1416 | 200 gon |
A handful of constants come up so often in triangle work that they deserve their own little table:
| Constant | Value | Where it appears |
|---|---|---|
| π | 3.14159265 | Radian measure; angle sum of a triangle in radians |
| √2 | 1.41421356 | Hypotenuse of the 45-45-90 triangle |
| √3 | 1.73205081 | Long leg of the 30-60-90 triangle; equilateral heights |
| 1 radian | 57.2957795° | Radians to degrees |
| 1 degree | 0.01745329 rad | Degrees to radians |
| 1 gon | 0.9° exactly | Surveying angles to degrees |
Length Units the Calculator Supports
A triangle does not care whether you measure it in millimeters or miles — its angles and proportions are identical at any scale — so the calculator lets you pick the unit that matches your tape measure and labels every result accordingly, including squared units for the area (enter sides in cm, read the area in cm²). All eight supported units, with their exact definitions:
| Unit | Symbol | Exact value | Typical use |
|---|---|---|---|
| Millimeter | mm | 0.001 m | Machining, 3D printing, technical drawing |
| Centimeter | cm | 0.01 m | Schoolwork, crafts, sewing |
| Meter | m | SI base unit | Construction, rooms, land |
| Kilometer | km | 1000 m | Maps, navigation, large surveys |
| Inch | in | 2.54 cm exactly | Woodworking, screens, US and UK trades |
| Foot | ft | 12 in = 0.3048 m | Building plans, room dimensions |
| Yard | yd | 3 ft = 0.9144 m | Fabric, sports fields, landscaping |
| Mile | mi | 1760 yd = 1.609344 km | Road distances, long baselines |
Special Triangles Worth Memorizing
A few triangles appear so often — on job sites, in exams, inside other problems — that knowing them by heart pays off for a lifetime. All of them are one-second jobs for the calculator, which also makes them perfect for testing that you and the tool agree.
The 3-4-5 Triangle and Other Pythagorean Triples
Sides of 3, 4 and 5 always form a perfect right angle, because 3² + 4² = 9 + 16 = 25 = 5². Builders have exploited this for millennia — the story goes that Egyptian rope-stretchers carried cords knotted into twelve equal segments for exactly this purpose — and the trick still squares deck frames and foundation corners today: measure 3 units along one wall, 4 along the other, and adjust until the diagonal is exactly 5. Any whole-number multiple works identically: 6-8-10, 9-12-15, 30-40-50. Sets of whole numbers satisfying a² + b² = c² are called Pythagorean triples, and the small ones are worth recognizing on sight:
| Triple (a, b, c) | Check | Non-right angles |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 36.87° and 53.13° |
| 5, 12, 13 | 25 + 144 = 169 | 22.62° and 67.38° |
| 8, 15, 17 | 64 + 225 = 289 | 28.07° and 61.93° |
| 7, 24, 25 | 49 + 576 = 625 | 16.26° and 73.74° |
| 20, 21, 29 | 400 + 441 = 841 | 43.60° and 46.40° |
| 9, 40, 41 | 81 + 1600 = 1681 | 12.68° and 77.32° |
The 30-60-90 and 45-45-90 Triangles
Two right triangles come with side ratios you can carry in your head. Slice an equilateral triangle down the middle and you get the 30-60-90; slice a square along its diagonal and you get the 45-45-90. Between them they answer a remarkable share of everyday geometry questions without any calculator at all — though checking yourself against this one takes two seconds.
| Triangle | Side ratio | Example | Remember it as |
|---|---|---|---|
| 30-60-90 | 1 : √3 : 2 | 5, 8.660, 10 | Short leg, short leg × √3, short leg × 2 |
| 45-45-90 | 1 : 1 : √2 | 5, 5, 7.071 | Two equal legs; hypotenuse = leg × √2 |
Equilateral Triangle Quick Formulas
The equilateral triangle is geometry’s minimalist masterpiece: one number — the side length a — determines absolutely everything about it. These shortcuts are exactly what the calculator computes when you enter three equal sides, condensed into a pocket card:
| Quantity | Exact formula | Decimal shortcut |
|---|---|---|
| Every angle | 60° | π / 3 ≈ 1.0472 rad |
| Height | (√3 / 2) × a | ≈ 0.8660 × a |
| Area | (√3 / 4) × a² | ≈ 0.4330 × a² |
| Perimeter | 3a | — |
| Circumradius R | a / √3 | ≈ 0.5774 × a |
| Inradius r | a / (2√3) | ≈ 0.2887 × a (exactly half of R) |
How to Use the Triangle Calculator, Step by Step
The tool was designed so that the first-time visitor and the daily user move at the same speed. If you like instructions anyway, here is the complete tour:
- Pick your mode. Choose the tab matching what you already know: SSS, SAS, ASA, AAS or SSA. The hint line beneath the tabs spells out exactly which three values that mode expects, in plain language.
- Set your units. Choose a length unit (mm through mi), an angle unit (degrees, radians or gradians) and, if you like, a fixed number of decimal places from 0 to 8 — or leave it on Auto for clean, six-significant-figure results with no trailing clutter.
- Enter your three values. The input boxes are forgiving by design: they accept a comma as the decimal separator (3,5 means 3.5), ignore stray spaces, and even read Eastern Arabic and Devanagari digits — type ٣٫٥ and the tool understands 3.5.
- Calculate. Click the button or simply press Enter. Invalid inputs get a specific, human explanation — sides that violate the triangle inequality, angles outside the 0–180° range, angle pairs that already exceed 180° — never a silent wrong answer.
- Read the drawing first. Every solution is rendered to true scale on a graph-paper grid, with vertices A, B, C labeled, each side tagged with its length, each corner tagged with its angle — and if one angle is exactly 90°, the classic little square marker appears there automatically. One glance tells you whether the shape matches your mental picture.
- Open More measurements. Heights, medians, angle bisectors, semi-perimeter, inradius and circumradius wait one tap away, grouped and labeled with the same h(a), m(a), t(a) notation used throughout this article.
- Copy, or go fullscreen. The Copy results button places a clean text summary on your clipboard, ready for homework, an email or a site log. The fullscreen button expands the calculator to fill the display — ideal for classrooms and workshop walls — and gracefully falls back to a built-in expanded view on devices, like iPhones, that restrict native fullscreen.
Three quieter conveniences round it out. An Example button fills each mode with sensible sample values, so you can watch the tool work before trusting it with your own numbers. Switching the angle unit converts any values you have already typed, in place, so nothing is lost to a settings change. And your language, units and decimals are remembered for your next visit — on your own device, by your own browser, with nothing sent anywhere.
Worked Examples You Can Reproduce in Seconds
Numbers persuade better than promises. Each example below states the inputs, the mode, and the results you should see — try any of them in the calculator above and watch the figures match.
Example 1 — Three Sides (SSS): the Famous 3-4-5
Enter a = 3, b = 4, c = 5 in SSS mode. The calculator reports angles of 36.8699°, 53.1301° and exactly 90°, confirms the type as Scalene • Right, and draws the right-angle marker at vertex C. The area is exactly 6, the perimeter 12, the semi-perimeter 6. Open the extra measurements and the elegance continues: the inradius is exactly 1 (for any right triangle, r = (a + b − c) / 2, and here that is (3 + 4 − 5) / 2), the circumradius is 2.5 — half the hypotenuse, because a right triangle’s circumcenter sits at the hypotenuse’s midpoint — and the median to the hypotenuse, m(c), is also 2.5 for exactly the same reason. The height onto the hypotenuse is 2.4, which is just 2 × area ÷ 5. One humble triple, five little theorems illustrated at once.
Example 2 — Two Sides and the Included Angle (SAS): 8, 5 and 60°
Enter b = 8, c = 5 and the included angle A = 60°. The law of cosines gives the missing side a as √(64 + 25 − 2 × 8 × 5 × 0.5) = √49 = exactly 7 — one of those rare textbook-perfect outcomes. The remaining angles come out as B ≈ 81.79° and C ≈ 38.21°, summing with the 60° to a clean 180°. The area is 10√3 ≈ 17.3205 (either from ½ × 8 × 5 × sin 60° or from Heron’s formula with s = 10 — both roads meet), the perimeter is 20, the inradius is √3 ≈ 1.7321, and the circumradius is about 4.0415.
Example 3 — Measuring Across a River (ASA)
You are standing on a riverbank and need the distance to a tree on the far side — without a boat. Plant two stakes 100 m apart along your bank (that is side a, your baseline). From the first stake, the angle between the baseline and the tree measures 65°; from the second stake, 40°. Enter B = 65°, a = 100, C = 40° in ASA mode. The tool finds the third angle A = 75°, then the two sightlines: about 66.55 m from the first stake to the tree and 93.83 m from the second. Better still, the perpendicular width of the river is the triangle’s height onto the baseline, h(a) ≈ 60.32 m — read straight from the More measurements panel. Two angle readings and one tape measurement, and the river surrenders its width. This, at industrial precision, is how surveying has worked for centuries.
Example 4 — The Ambiguous Case in Action (SSA): 6, 8 and 30°
Enter a = 6, b = 8 and A = 30° in SSA mode. Check the condition first: b × sin A = 8 × 0.5 = 4, and since 4 < 6 < 8, geometry promises two triangles — and the calculator delivers both, drawn side by side. Solution 1: B ≈ 41.81°, C ≈ 108.19°, c ≈ 11.40, area ≈ 22.80, perimeter ≈ 25.40 — a wide, obtuse-topped shape. Solution 2: B ≈ 138.19°, C ≈ 11.81°, c ≈ 2.46, area ≈ 4.91, perimeter ≈ 16.46 — a slender sliver of a triangle. Same three inputs, two completely legitimate answers, and the note above the results explains why. If a homework problem ever asks for “all possible triangles,” this is the trap it is setting.
Where Triangle Math Earns Its Keep in Real Life
It is tempting to file trigonometry under “things I survived in school,” but triangle solving is quietly load-bearing across an astonishing range of work.
Construction and carpentry. Squaring a foundation or a deck frame is the 3-4-5 check performed with a tape measure. Rafters are right triangles: run along the ceiling joist, rise to the ridge, rafter as hypotenuse — SSS or SAS mode answers cut lengths and pitch angles before the saw comes out. Stair stringers, gussets, braces and shelf brackets are all the same triangle wearing different clothes.
Surveying and mapping. Triangulation — covering territory with a network of triangles solved one baseline at a time — mapped entire nations long before satellites, and the method survives inside every modern total station. It is also why the gradian unit lives in this calculator: surveying instruments across much of the world still read in gon, where a right angle is a tidy 100.
Navigation. Take bearings to a lighthouse from two points along your course and you have an ASA problem whose solution is your distance offshore. Aviators, sailors and hikers with a compass have run this exact triangle for generations; your phone’s positioning does a mathematically fancier cousin of it many times per second.
Land and area measurement. Any straight-sided parcel, however irregular, splits into triangles; measure each triangle’s three sides, let Heron’s formula produce each area, and add. It is the honest, checkable way to compute acreage with nothing but a measuring wheel.
Design, fabrication and 3D printing. Brackets, jigs, trusses and frames all reduce to triangles at dimensioning time, and CAD software leans on the same laws of sines and cosines this page describes. Checking a bracket’s geometry here before committing filament or steel is a cheap habit with expensive-mistake insurance built in.
Physics and engineering coursework. Resolving forces on an inclined plane, adding vectors tip-to-tail, computing a ladder’s safe lean — each is a triangle problem in costume. The calculator will not write your lab report, but it will confirm every triangle inside it.
Astronomy. The distances to nearby stars were first measured by parallax: Earth’s orbit as a 300-million-kilometer baseline, two angle sightings six months apart, one very long skinny triangle. The same ASA logic that crossed our river in Example 3, stretched across light-years — and precisely the regime where numerically stable formulas stop being a luxury.
Accuracy, Privacy and Speed
A calculator earns trust three ways, and this one was built for all three. Accuracy: every result comes from the stable formulations described above — Kahan’s area rearrangement, half-angle arctangent recovery of angles, and a final pass that guarantees the three reported angles sum to exactly 180°. Before publication the engine passed a suite of more than one hundred automated tests against triangles with known exact answers, from the 3-4-5 through the ambiguous SSA pairs to deliberately pathological needle triangles with sides a billion times apart.
Privacy: everything you type is processed by your own browser on your own device. No side length, no angle, no result is transmitted, stored or logged — there is nothing to transmit it to. Close the tab and the session is gone. Speed: because there is no server round-trip and no external file to fetch, results appear the instant you press Calculate, on a five-year-old phone as readily as on a workstation. The tool also respects your device’s reduced-motion preference, keeping the experience comfortable for everyone.
Common Mistakes and How to Avoid Them
Years of triangle questions repeat the same handful of stumbles. Skim this list once and you will dodge nearly all of them:
- Wrong angle unit. Typing 45 while the selector says radians describes an angle of about 2578° worth of turns — not a corner. Glance at the little unit badge inside each input box before calculating; it always shows the active unit.
- Ignoring the triangle inequality. If two sides do not add to more than the third, no triangle exists — the tool will say so rather than improvise. When inventing practice numbers, keep the longest side shorter than the other two combined.
- Expecting one answer from SSA. Two sides and a non-included angle can legitimately produce two triangles. If the calculator shows Solution 1 and Solution 2, that is geometry being thorough, not the tool being indecisive.
- Rounding too early by hand. Carrying 36.87° forward as 37° through three more steps quietly bends every later result. The calculator keeps full precision internally and rounds only for display — do the same on paper, or let the Decimals setting round for you at the very end.
- Confusing included and non-included. SAS wants the angle between the two sides; SSA wants an angle opposite one of them. The hint line under the mode tabs states which is which every time — read it once and the distinction sticks.
- Trying to solve from three angles. AAA fixes shape, not size; a triangle with angles 40°-60°-80° can be any size at all. You always need at least one side.
- Entering 0° or 180° as an angle. Both flatten the triangle into a line segment. Valid interior angles live strictly between the two.
Frequently Asked Questions
How do I find the missing side of a triangle?
It depends on what you know. With two sides and the angle between them, the law of cosines gives the third side directly: c² = a² + b² − 2ab cos C. With one side and two angles, find the third angle (they must total 180°) and scale the other sides with the law of sines. In the calculator, that is SAS mode for the first situation and ASA or AAS for the second — no formula handling required.
How do I find a missing angle?
If you already know two angles, subtract their sum from 180°. If you know all three sides, the rearranged law of cosines recovers any angle: cos A = (b² + c² − a²) / (2bc). SSS mode performs this — through an even more precise equivalent — for all three angles at once.
Why do a triangle’s angles always add up to 180 degrees?
Draw a line through one vertex parallel to the opposite side; the three angles of the triangle reassemble along that line as a straight angle. It is one of the oldest results in geometry, and it holds for every triangle on a flat surface — equivalently π radians or 200 gradians. (On curved surfaces like a sphere the sum exceeds 180°, which is a story for another article.)
Can SSA really give two different triangles?
Yes. When the given angle A is acute and the side opposite it satisfies b sin A < a < b, two distinct triangles fit the same three values — one with an acute angle at B, one with its obtuse supplement. This calculator detects the situation and displays both solutions in full, each with its own scale drawing.
Is a 3-4-5 triangle always a right triangle?
Always — because 3² + 4² = 5² satisfies the Pythagorean theorem exactly, and the converse of that theorem guarantees the right angle. Every scaled copy (6-8-10, 30-40-50, 1.5-2-2.5) is right-angled too, which is precisely what makes the string trick so useful on a job site.
How do I convert degrees to radians or gradians?
Multiply degrees by π/180 (≈ 0.0174533) to get radians, or by 10/9 to get gradians. Going the other way, multiply radians by 180/π (≈ 57.2958) or gradians by 0.9. The calculator does this live: switch the angle-unit selector and any value you have typed is converted in place.
What is the difference between the law of sines and the law of cosines?
The law of cosines relates three sides and one angle, so it shines when your knowns are side-heavy (SSS and SAS). The law of sines relates side-angle opposite pairs, so it shines when your knowns are angle-heavy (ASA, AAS and SSA). Between the two, every solvable triangle configuration is covered — which is exactly how this calculator divides the work internally.
Can I calculate the area of a triangle without knowing its height?
Yes, two ways. From three sides, Heron’s formula: area = √(s(s−a)(s−b)(s−c)) with s the semi-perimeter. From two sides and their included angle: area = ½ ab sin C. The tool applies whichever fits your input — and then reports the three heights anyway, computed backwards from the area as h = 2 × area ÷ base.
What units does this triangle calculator support?
Eight length units — millimeters, centimeters, meters, kilometers, inches, feet, yards and miles, with areas in the matching squared unit — and three angle units: degrees, radians and gradians. Input is flexible too: comma decimals and Eastern Arabic or Devanagari digits are all understood.
Is the calculator free, and what happens to my data?
Completely free, with no account, no ads and no usage limits. All computation happens locally in your browser; nothing you enter is transmitted, stored or logged anywhere. Your triangles are your business.
Triangle Calculator Embed Code for Your Website
Everything you have just read — the five input methods, the law of sines and law of cosines working in tandem, the honest handling of the ambiguous SSA case, the stable area mathematics, the to-scale drawing, the six languages — lives inside one self-contained tool, and you can put that exact tool on your own website. The embed below is the full calculator, not a preview: your visitors solve triangles, read areas and perimeters, open the extended measurements and see every diagram without ever leaving your page.
It behaves like a well-mannered guest. The calculator runs entirely in the visitor’s browser, inside an isolated container that cannot read or alter the page around it. No side lengths, no angles, no results are transmitted, stored or logged — every calculation happens locally, on the visitor’s own device, using the same high-precision arithmetic described earlier in this article. There are no server round-trips to wait for and nothing to break when traffic spikes. Close the tab and the whole session evaporates, which is exactly how a geometry tool should behave.
The embed ships in six interface languages — English, Spanish, French, Arabic, Chinese and Hindi — covering the labels, buttons, hints, error messages and result panels alike. The language is detected automatically from each visitor’s browser on first visit and remembered afterward, so the same embed serves a worldwide audience without any configuration on your part. For Arabic, the entire layout mirrors into right-to-left so the experience feels native rather than translated.
Prefer to fix the language yourself? Set it with a URL parameter: ?lang=en
Available languages: en — es — fr — ar — zh — hi
1- Isolated iFrame Code (Ideal for Articles and Pages):
This is the method to reach for when the calculator belongs in the flow of your content — a geometry guide, a trigonometry lesson, a resource page for students, teachers, architects or engineers, exactly like the article you are reading now. The isolated iframe keeps the tool neatly contained: your layout stays untouched, your styles stay yours, and the embedded calculator resizes itself automatically to fit its content. Paste the two blocks below into your page’s HTML and the full tool appears, ready to use.
<iframe data-tooliqo src="https://tools.tooliqo.co/triangle-calculator/?lang=en"
title="Tooliqo — triangle-calculator"
style="width:100%;border:0;height:900px" height="900" loading="lazy" scrolling="no"
allowfullscreen allow="fullscreen; clipboard-write"></iframe>
<script src="https://tools.tooliqo.co/embed.js" async></script>
2- Flexible Script (Suitable for Sidebars and Templates):
For sidebars, footers, widget areas and theme templates, this lighter integration is the better fit. It loads the calculator on demand and adapts to whatever container you give it — the same engine, the same five input methods, the same measurements, drawings, validation and six languages, simply packaged for the places where a fixed iframe would feel heavy. Drop the snippet where you want the tool to appear, and it appears.
<div class="tooliqo-tool" data-tool="triangle-calculator" data-lang="en"data-height="900"></div><script src="https://tools.tooliqo.co/embed.js" async></script>
Note: Both methods load the identical calculator described throughout this article — same input methods, same solving engines, same area, perimeter and height calculations, same diagrams, same validation, same six languages. The tool respects your visitors’ dark-mode and reduced-motion settings, and comes with no watermarks, no daily limits and no data collection of any kind. It simply does its job on your site. And if one visitor enters three side lengths, watches the triangle take shape on screen, and finally understands why a triangle with sides 3, 4 and 5 must be right-angled — then the tool has done what it was built to do. The rest is up to them, which was always the point.
A Last Word
The triangle has been humanity’s favorite problem-solving machine for at least four thousand years — from rope-stretchers squaring temple foundations, to surveyors triangulating continents, to astronomers hanging distance markers on the stars. The mathematics never changed; only the tools did. This calculator is simply the latest tool in that long line: the law of sines, the law of cosines and Heron’s formula, executed carefully, drawn honestly, translated into six languages, and handed to you free of charge and free of surveillance. Bookmark the page, keep the conversion tables above within reach, and the next time a triangle stands between you and an answer — on a worksheet, a job site or a riverbank — you will have it solved before the coffee cools.