Triangle Calculator | Solve Sides, Angles & Area Instantly (SSS/SAS/ASA)
Enter three sides, or two sides and an angle, and instantly get the remaining sides, angles, area, perimeter, circumradius, and inradius — free, with SSS, SAS, and ASA input modes. Also checks whether your triangle is valid.
What Is Triangle Calculation
A triangle is fully determined the moment you know three of its six measurements — its sides and angles. This triangle calculator covers the three most common ways of supplying that information: SSS (three sides), SAS (two sides and the angle between them), and ASA (one side and the two angles next to it). Enter whichever combination you have, and the tool works out the remaining sides and angles, then goes further to compute the area, perimeter, circumradius, and inradius in a single pass. It also classifies the triangle two ways at once — by its angles (acute, right, or obtuse) and by its sides (equilateral, isosceles, or scalene) — so you get a complete picture of the shape you're working with, not just a list of numbers.
The calculator works with triangles on a flat plane. In SSS mode, it automatically checks the triangle inequality — the rule that any two sides must add up to more than the third — and if your values fail that test, it tells you the triangle can't exist rather than returning a misleading result. Angles are always entered and displayed in degrees. Side lengths carry no fixed unit: whatever unit you use (meters, inches, or anything else) flows straight through to the area, perimeter, and both radii, so the output is ready to use without any manual unit conversion.
How to Use the Triangle Calculator
- Pick the type of information you have Choose SSS if you know all three sides, SAS if you know two sides and the angle between them, or ASA if you know one side and the angles at both of its ends.
- Enter the values The input fields change to match the mode you picked. Double-check the labels as you type, especially in SAS mode — plugging in the wrong angle produces a completely different triangle.
- Review the results and classification The remaining sides and angles appear instantly, along with area, perimeter, circumradius, inradius, and the triangle's classification by angle and by side.
- Adjust your input if the triangle isn't valid If you see "not a valid triangle," check whether your side lengths satisfy the triangle inequality, or whether your angles add up to less than 180° in ASA mode.
- Cross-check with a different mode when possible If you happen to have more than three known values, entering the same triangle a second time through a different mode (say, SSS instead of SAS) is a quick way to confirm your measurements are consistent.
Tips for getting more out of it
- In SSS mode, values that violate the triangle inequality (any two sides must sum to more than the third) will show "not a valid triangle."
- Whether a triangle is a right triangle is determined by checking if its largest angle is 90°, which also works as a quick check for Pythagorean triples like 3-4-5.
- The circumradius is derived from the law of sines (a / (2 sin A)); the inradius comes from area divided by the semi-perimeter.
- SAS and ASA modes are handy for checking surveying or geometry problems where only two pieces of information are known.
Ways to Use Triangle Calculation
Checking homework and exam prep
If you worked out the law of cosines or law of sines by hand, enter the same starting values here and compare answers. It's a fast way to narrow down exactly where an arithmetic slip happened, instead of re-deriving the whole problem.
Finding the area of a plot of land
If a piece of land can be broken down into triangles, three measured side lengths are enough to calculate its area. An irregular quadrilateral can likewise be split along a diagonal into two triangles and measured piece by piece.
Working out dimensions for a build
For a diagonal brace on a shelf or an angled piece of framing, you often only know two sides and the angle between them. SAS mode gives you the missing third side before you cut any material.
Verifying that a corner is actually square
Measure the three sides of a frame or corner you've built and enter them in SSS mode. If the largest angle comes out to exactly 90°, the corner is genuinely square — a numeric check that beats eyeballing it.
Working through geometry proofs and constructions
When a proof involves the circumcircle, incircle, or a congruence condition, plugging in concrete numbers and comparing the calculated radii against your own construction makes the abstract result much easier to grasp.
Triangle Calculation Glossary
- Triangle inequality
- The rule that the sum of any two side lengths must be greater than the third side. Three lengths that fail this test cannot form a triangle on a flat plane, which is why the calculator reports "not a valid triangle" for them.
- Law of cosines
- A formula for finding a triangle's angles from its three side lengths (cos A = (b² + c² − a²) / 2bc). It's also what determines the third side when you know two sides and the included angle.
- Law of sines
- The relationship stating that the ratio of a side's length to the sine of its opposite angle is the same for all three sides of a triangle. The circumradius can be derived directly from this relationship as well.
- Circumcircle
- The circle that passes through all three vertices of a triangle. Its center sits at the point where the perpendicular bisectors of the three sides meet, equidistant from every vertex.
- Incircle
- The circle that touches all three sides of a triangle from the inside. Its radius equals the triangle's area divided by half its perimeter, and its center lies where the angle bisectors meet.
- Included angle
- The angle that sits directly between two specified sides. This is the angle SAS mode expects — entering a different angle of the triangle instead produces an entirely different shape.
- Congruence conditions
- The rules (SSS, SAS, ASA, and a few others) that guarantee two triangles are identical in shape and size whenever certain sides and angles match. They explain why just three well-chosen measurements are enough to pin a triangle down uniquely.
Frequently Asked Questions
Side Note — Why SSS, SAS, and ASA are enough to "fix" a triangle
The triangle congruence conditions taught in school — three sides, two sides and the included angle, one side and its two adjacent angles — really describe the minimum amount of information needed to pin down a triangle uniquely. Fixing three points in a plane would normally take six degrees of freedom (each point's x and y coordinates), but once you remove the three degrees of freedom for translation, rotation, and reflection, only three independent pieces of information — some combination of sides and angles — are left to determine the shape. SSS, SAS, and ASA are the standard ways of supplying exactly that.
By contrast, two angles plus a side that isn't adjacent to both of them (often called SSA) can be ambiguous — the same two pieces of information can sometimes correspond to two different triangles. This "ambiguous case" trips up a lot of students in the law-of-sines unit of a geometry course precisely because it looks so similar to the SAS condition that actually does determine a triangle uniquely.
The circumcircle and incircle are also consequences of a triangle being fully determined by its "three pieces of information." Any three non-collinear points determine exactly one circle passing through them, and its center can be constructed as the intersection of the perpendicular bisectors of the sides. This same geometric fact underpinned pre-GPS surveying: triangulation lets you locate an unknown point purely from two known points and a couple of measured angles.