The Triangle Calculator solves a triangle from any three of its six measurements, its three sides and three angles, and returns the remaining values, the area, the perimeter and a labelled diagram drawn to the entered dimensions. It names the case it detected, shows which theorem solved each step, and handles the ambiguous case where two different triangles fit the same measurements.
Three measurements fix a triangle, with one important exception covered below. Enter what you know and the calculator finds the rest, showing the working so the result can be checked by hand.
Solve a triangle from any three measurements
A triangle has six measurements: three side lengths and three interior angles. Knowing three of them, as long as at least one is a side, determines the whole triangle. That is why every triangle problem gives you exactly three pieces of information.
Enter any valid combination: three sides, two sides and an angle, or two angles and a side. The calculator identifies which combination it has, chooses the method that fits, and solves for the other three measurements. It also returns the area, perimeter, the three heights, the medians, and the radii of the inscribed and circumscribed circles.
Each measurement carries its own unit, and angles can be entered in degrees or radians. The diagram redraws to match the numbers, with every side and angle labelled, so the shape is visible rather than abstract.
Identify which case you have
Which method solves a triangle depends on which three measurements are known. Mathematicians name these cases by their pattern of sides (S) and angles (A), and identifying the case is the first step in every solution.
| Case | You know | Method |
|---|---|---|
| SSS | three sides | law of cosines |
| SAS | two sides and the angle between them | law of cosines, then law of sines |
| ASA | two angles and the side between them | angle sum, then law of sines |
| AAS | two angles and a side not between them | angle sum, then law of sines |
| SSA | two sides and an angle not between them | law of sines, the ambiguous case |
The letters record the order the measurements sit around the triangle. SAS means the known angle is *between* the two known sides; SSA means it is not. That difference matters more than it looks, because SSA is the one case that can have two answers.
The calculator detects the case automatically and names it, so the method is never a guess.
Solve with the law of sines
The law of sines relates each side to the sine of its opposite angle, and it solves the cases where a side and its opposite angle are both known, ASA, AAS, and SSA.
a / sin(A) = b / sin(B) = c / sin(C)Every side divided by the sine of the angle facing it gives the same ratio. So knowing one side and its opposite angle sets the ratio for the whole triangle, and any other known angle then gives its opposite side.
For an ASA or AAS triangle, the third angle comes first from the angle sum, all three interior angles total 180 degrees, and then the law of sines finds the two unknown sides. The method is direct and gives a single answer.
Solve with the law of cosines
The law of cosines generalises the Pythagorean theorem to any triangle, and it solves the cases where the law of sines cannot start: SSS and SAS.
c² = a² + b² − 2ab·cos(C)When all three sides are known (SSS), the formula rearranges to find each angle:
cos(C) = (a² + b² − c²) / 2abWhen two sides and the included angle are known (SAS), the formula finds the third side directly, after which the law of sines finishes the angles. Notice that when the angle C is 90 degrees, cos(C) is zero and the formula collapses to a² + b² = c², the law of cosines contains the Pythagorean theorem as its right-angled special case.
Handle the ambiguous SSA case
SSA is the case that trips up both students and calculators, and it is where the Triangle Calculator does something most competitors do not: it returns every valid triangle, not just the first one it finds.
When you know two sides and an angle that is not between them, the constraints can be satisfied by two different triangles, one triangle, or none at all.
The reason is geometric: the known side swings like a hinge, and depending on its length it can meet the opposite baseline at two points, one point, or not reach it.
Consider two sides and a non-included angle where the side opposite the known angle is shorter than the other known side but long enough to reach the base. Two triangles close that gap, one where the third angle is acute, one where it is obtuse. Both are valid. A calculator that reports only one is giving half the answer, and a student who wrote only one on an exam would lose marks.
The Triangle Calculator checks for this. When two triangles exist, it draws both and lists both sets of measurements side by side. When the configuration is impossible, the swinging side is too short to reach, it says so plainly instead of returning a wrong number. When the answer is unique, it confirms why.
This is the single most useful thing the calculator does, because the ambiguous case is exactly where triangle problems go wrong.
Calculate the area of a triangle
The area of a triangle can be found three ways, and the calculator picks whichever the known measurements allow. From a base and its height: `` A = ½ × base × height ``
From two sides and the included angle (SAS): `` A = ½ × a × b × sin(C) ``
From all three sides, using Heron's formula (SSS): `` A = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) / 2 ``
Heron's formula is the one to reach for when only the sides are known and no height is given. The term s is the semi-perimeter, half the sum of the sides. For a triangle with sides 6, 8 and 10, the semi-perimeter is 12, and the area works out to √(12 × 6 × 4 × 2) = √576 = 24.
Solve equilateral and isosceles triangles
Equilateral and isosceles triangles are special cases the calculator solves from fewer inputs, because their equal sides supply the missing information. An equilateral triangle has three equal sides and three 60-degree angles. A single side length fixes everything: the area is (√3 / 4) × a² and the height is (√3 / 2) × a.
Enter one number and the whole triangle follows.
An isosceles triangle has two equal sides and two equal base angles. Knowing the equal side and the base, or the base and one angle, is enough to solve it. Its area, with equal legs a and base b, is (b/4) × √(4a² − b²).
Both are modes of the same solver, the calculator applies the equal-side constraint and needs less input as a result.
Solve a triangle with sides 6, 8 and 10
Take the SSS case with sides 6, 8 and 10, which turns out to be a right triangle. 1. Identify the case. Three sides are known, so this is SSS and the law of cosines finds the angles. 2. Find the largest angle first. The largest angle faces the longest side, 10.
Using the law of cosines: `` cos(C) = (6² + 8² − 10²) / (2 × 6 × 8) = (36 + 64 − 100) / 96 = 0 / 96 = 0 `` cos(C) = 0 means C = 90 degrees. The triangle is right-angled, which the sides 6-8-10, a scaled 3-4-5, confirm.
3. Find a second angle. `` cos(A) = (8² + 10² − 6²) / (2 × 8 × 10) = (64 + 100 − 36) / 160 = 128 / 160 = 0.8 `` A = 36.87 degrees.
4. Find the third by the angle sum. B = 180 − 90 − 36.87 = 53.13 degrees.
5. Find the area with Heron's formula. s = (6 + 8 + 10) / 2 = 12, so A = √(12 × 6 × 4 × 2) = √576 = 24.
Every step is shown by the calculator, and the diagram marks the right angle with a square so the result is visible at a glance.
Frequently asked questions
How do I solve a triangle?
To solve a triangle, enter any three of its six measurements, including at least one side, into the Triangle Calculator. The calculator identifies the case, SSS, SAS, ASA, AAS or SSA, chooses the law of sines or law of cosines accordingly, and returns the remaining sides and angles, the area, and the perimeter, with each step shown.
What is the SSA ambiguous case?
The SSA ambiguous case arises when two sides and a non-included angle can be satisfied by two different triangles. The side opposite the known angle can swing to meet the base at two points, giving one acute and one obtuse solution. The Triangle Calculator returns both valid triangles, or states clearly when the configuration allows only one or none.
When do I use the law of sines versus the law of cosines?
Use the law of sines when a side and its opposite angle are both known, which covers the ASA, AAS and SSA cases. Use the law of cosines when they are not, the SSS case, where three sides are known, and the SAS case, where two sides and the angle between them are known. The calculator selects the correct law automatically.
How do I find the area of a triangle without the height?
When the height is unknown, use Heron's formula, which finds the area from the three side lengths alone: A = √(s(s−a)(s−b)(s−c)), where s is half the perimeter. If two sides and their included angle are known instead, the area is ½ × a × b × sin(C). The Triangle Calculator applies whichever the known measurements allow.
Can any three measurements solve a triangle?
Almost. Any three measurements that include at least one side will solve a triangle, because three angles alone fix the shape but not the size, infinitely many triangles share the same angles. The one complication is the SSA case, where three valid measurements can still describe two different triangles.
Why does a triangle sometimes have two valid answers?
A triangle has two valid answers only in the SSA case, where two sides and a non-included angle are known. The side opposite the given angle can reach the base at two different points, forming one triangle with an acute angle and another with an obtuse angle. Both satisfy the measurements, so both are correct.
What is Heron's formula?
Heron's formula calculates a triangle's area from its three side lengths without needing the height. It states that the area equals √(s(s−a)(s−b)(s−c)), where a, b and c are the sides and s is the semi-perimeter, (a+b+c)/2. It is the standard method for the SSS case and is named after Hero of Alexandria.
Do the three angles always add up to 180 degrees?
Yes. In any flat, or Euclidean, triangle the three interior angles always sum to exactly 180 degrees. This fact lets the calculator find a third angle immediately whenever two are known, which is the first step in solving the ASA and AAS cases. The rule does not hold on curved surfaces, but standard triangle problems assume a flat plane.
Summary
The Triangle Calculator solves any triangle from three known measurements, provided at least one is a side, and returns every remaining side and angle along with the area, perimeter and a labelled diagram.
It detects the case, SSS, SAS, ASA, AAS or SSA, and applies the law of cosines for known sides or the law of sines when a side and its opposite angle are paired.
Its most important feature is correct handling of the ambiguous SSA case, where two sides and a non-included angle can describe two valid triangles: the calculator returns both, rather than the single answer most tools give. Areas come from base-and-height, the two-sides-and-angle formula, or Heron's formula for three known sides, and equilateral and isosceles triangles solve from fewer inputs thanks to their equal sides.