Triangle Calculator with Steps
Enter any valid set of three measurements, including at least one side. Solve SSS, SAS, ASA, AAS and SSA triangles, then find missing sides, angles, perimeter, area, altitudes, medians, angle bisectors and circle radii.
Last Updated: July 26, 2026Online Triangle Solver
Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Leave unknown measurements blank.
Solved Triangle
Enter measurements, then select Solve Triangle.
Sides and opposite angles
| Pair | Side length | Angle | Altitude |
|---|---|---|---|
| Triangle measurements will appear here. | |||
Additional measurements
| Measure | At A | At B | At C |
|---|---|---|---|
| Medians and angle bisectors will appear here. | |||
Circle and area measures
| Semiperimeter | Inradius | Circumradius | Area |
|---|---|---|---|
| Derived values will appear here. | |||
Calculation steps
- Enter three known measurements, including at least one side.
How to Use This Triangle Calculator
- Match each side with its opposite angle: a with A, b with B, and c with C.
- Enter at least three known measurements. One of them must be a side because three angles define only a shape, not its scale.
- Leave every unknown field blank. Use the same unit for all side lengths.
- Select a length label and the number of decimal places you want to display.
- Select Solve Triangle. Review each valid solution, formulas, measurements and the scaled diagram.
You can enter more than three values. The solver uses the extra measurements as checks. If they disagree with the triangle produced by another valid subset, it reports inconsistent data instead of choosing one answer silently. The quick examples demonstrate SSS, SAS, ASA and the two-solution SSA case.
What This Triangle Calculator Solves
A triangle has three side lengths and three interior angles. Six measurements describe it, but a valid combination of three usually determines the rest. This calculator identifies the input case, applies the Law of Sines, Law of Cosines or angle-sum rule, and then calculates the complete triangle.
The standard notation matters. Lowercase side a lies across from uppercase angle A. The same relationship applies to b and B, and to c and C. An angle entered beside the wrong side changes the problem type and can produce a different triangle.
| Known measurements | Case | Main method | Possible answers |
|---|---|---|---|
| Three sides | SSS | Law of Cosines | One or none |
| Two sides and included angle | SAS | Law of Cosines | One |
| Two angles and one side | ASA or AAS | Angle sum and Law of Sines | One |
| Two sides and a non-included angle | SSA | Law of Sines | Zero, one or two |
| Three angles only | AAA | Angle sum | Scale is unknown |
Triangle Formulas Used
Angle Sum
The three interior angles of a non-degenerate Euclidean triangle add to 180 degrees. When two angles are known, the third follows directly.
C = 180° − A − B
Law of Sines
The Law of Sines connects every side with the sine of its opposite angle. It solves ASA and AAS cases and helps analyze SSA input.
Law of Cosines
The Law of Cosines generalizes the Pythagorean theorem. It solves SAS and SSS triangles. The formula shown for side a has matching versions for b and c.
cos(A) = (b² + c² − a²) / (2bc)
Heron's Formula for Area
Once all three sides are known, the semiperimeter s and Heron's formula give the area K. The calculator uses a numerically stable rearrangement of this formula internally.
K = √(s(s − a)(s − b)(s − c))
Worked Example: Solve a 3-4-5 Triangle
Enter a = 3, b = 4 and c = 5. This is an SSS problem. The sides pass the triangle inequality because each pair adds to more than the remaining side. Use the Law of Cosines to find angle A:
cos(A) = 32 / 40 = 0.8
A = 36.869898°
The same process gives B = 53.130102 degrees and C = 90 degrees. The perimeter is 3 + 4 + 5 = 12. The semiperimeter is 6, so Heron's formula gives an area of 6 square units. Because the largest angle is 90 degrees and all sides differ, this is a scalene right triangle.
The inradius is K/s = 6/6 = 1. The circumradius is abc/(4K) = 60/24 = 2.5. The altitudes opposite a, b and c are 4, 3 and 2.4 units. These extra measurements provide useful checks without changing the original SSS solution.
Understanding the SSA Ambiguous Case
SSA means two sides and an angle opposite one of those sides are known. It is different from SAS because the entered angle is not between the two known sides. The sine of an angle has the same value for an acute angle and its supplementary obtuse angle. As a result, the same measurements can form two different triangles.
For example, enter A = 30 degrees, a = 5 and b = 6. The Law of Sines gives sin(B) = 6 sin(30 degrees)/5 = 0.6. One angle is about 36.869898 degrees. Its supplement is about 143.130102 degrees. Both leave a positive third angle, so both triangles are valid. The solution buttons let you inspect each complete result.
- If the sine ratio is greater than 1, no triangle exists.
- If the ratio produces only one angle that leaves a positive third angle, there is one triangle.
- If both an angle and its supplement fit below 180 degrees with the known angle, there are two triangles.
Additional Triangle Measurements
After solving the six main measurements, the calculator derives values used in geometry, construction layouts, surveying exercises and trigonometry problems.
| Measure | Formula at vertex A | Meaning |
|---|---|---|
| Altitude | ha = 2K/a | Perpendicular distance from A to the line containing side a. |
| Median | ma = ½√(2b² + 2c² − a²) | Segment from A to the midpoint of side a. |
| Angle bisector | la = 2bc cos(A/2)/(b + c) | Segment that divides angle A into two equal angles. |
| Inradius | r = K/s | Radius of the circle tangent to all three sides. |
| Circumradius | R = abc/(4K) | Radius of the circle through all three vertices. |
The formulas for vertices B and C follow by rotating the side and angle labels. Every length result uses the selected common unit. Area uses the square of that unit.
Triangle Classification
Triangles are classified by sides and by angles. An equilateral triangle has three equal sides and three 60-degree angles. An isosceles triangle has at least two equal sides. A scalene triangle has three different sides.
For angle type, an acute triangle has three angles below 90 degrees. A right triangle has one 90-degree angle. An obtuse triangle has one angle above 90 degrees. The calculator compares squared side lengths before it labels the angle type, which avoids relying on a rounded displayed angle.
c² = a² + b²: right
c² < a² + b²: acute
c² > a² + b²: obtuse
Triangle Inequality and Invalid Inputs
Three positive side lengths form a non-degenerate triangle only when the sum of every two sides is greater than the third side. The common shortcut is to sort the sides and check that the two smaller sides add to more than the largest.
The solver also rejects zero or negative sides, angles outside 0 to 180 degrees, two known angles whose sum reaches 180 degrees, nonnumeric text, division by zero in a fraction and combinations that conflict with extra entered measurements. Extremely near-degenerate triangles are rejected when floating-point arithmetic cannot support a stable area.
- Use one unit for all sides. Convert first if your source measurements mix feet, inches, metres or centimetres.
- Enter 1250 instead of 1,250 because commas are not accepted inside a field.
- Leave unknown values blank. Do not enter zero as a placeholder.
- Keep extra digits during the calculation and round only the final value.
Accuracy, Units and Calculator Limits
The solver keeps full browser precision during intermediate calculations and rounds only the displayed output. It normalizes side lengths before area, angle and derived-length calculations to reduce avoidable overflow and cancellation. Valid side inputs range from 1e-150 through 1e150.
Browser arithmetic follows IEEE 754 double-precision rules. Most ordinary geometry problems receive far more precision than their source measurements justify, but values near a mathematical boundary need care. A triangle that is almost flat, an SSA ratio almost equal to 1, or measurements rounded before entry can change the reported case.
The unit selector labels results; it does not convert numbers. If you enter side a in centimetres and side b in inches, the answer has no consistent physical meaning. For drawings, manufacturing, structural work, navigation or surveying, follow the required tolerance and independently verify source data and results.
Method and Review Basis
The Law of Sines, ASA, AAS and ambiguous SSA explanations follow OpenStax on non-right triangles and the Law of Sines. The SAS, SSS, Law of Cosines and Heron's formula sections follow OpenStax on the Law of Cosines. Right-triangle ratios align with OpenStax right-triangle trigonometry.
The page uses a strict number parser and never executes entered text as code. Calculations and the temporary history list remain in the current page. Reloading clears them.
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Frequently Asked Questions
What is a triangle calculator?
A triangle calculator solves missing sides and angles from a valid SSS, SAS, ASA, AAS or SSA input set. It can also calculate perimeter, area, altitudes, medians, angle bisectors, inradius, circumradius and triangle type.
What measurements do I need to solve a triangle?
Enter at least three independent measurements, including at least one side. Valid sets include three sides, two sides and an included angle, two angles and one side, or an SSA combination that may have zero, one or two solutions.
Why are three angles not enough?
Three angles determine the triangle's shape but not its size. Infinitely many similar triangles share the same angles. At least one side is required to establish the scale and calculate actual lengths, perimeter and area.
How do I label sides and angles?
Use matching opposite pairs. Side a is across from angle A, side b is across from angle B, and side c is across from angle C. Entering an angle beside the wrong opposite side changes the problem.
Why does SSA sometimes have two solutions?
Sine has the same positive value for an acute angle and its supplementary obtuse angle. When both angles leave a positive third angle, the same SSA measurements form two distinct triangles. Other SSA inputs form one or no triangle.
How is triangle area calculated?
After all sides are known, the calculator uses a stable form of Heron's formula with semiperimeter s = (a + b + c)/2. It may also be checked using K = one-half times two sides times the sine of their included angle.
How does the calculator identify a right triangle?
It compares the square of the longest side with the sum of the squares of the other two sides. Equality within a small numerical tolerance indicates a right triangle, a larger value indicates obtuse, and a smaller value indicates acute.
Can I enter fractions or scientific notation?
Yes. Each field accepts a positive decimal, scientific-notation value or one simple fraction such as 7/2. Do not enter commas, units, mixed numbers, formulas or more than one fraction slash.
Why does the calculator say my measurements conflict?
Extra measurements must match the triangle determined by a valid three-value subset. A conflict often comes from rounding, mixed units, an angle paired with the wrong opposite side, or one incorrectly copied value.
Is the triangle diagram drawn to scale?
The diagram preserves the calculated shape and is fitted into the available space for readability. Labels and point markers take visual space, so use the numeric tables rather than screen pixels for measurements.
Disclaimer: This calculator provides general mathematical results for learning, estimation and checking work. Verify inputs, units, formula assumptions, tolerances and required precision before using a result for academic, engineering, construction, surveying, financial or safety-critical decisions.