Triangle Solver: Sides, Angles, Area and Law of Sines

How to use it

Enter exactly three values: any three of the sides Side a, Side b, Side c and the angles Angle α, Angle β, Angle γ. Leave the other fields empty.
Read Number of triangles. Two triangles appear only in the SSA case.
Check the sides, angles, area, perimeter, heights and type in the table.
Go through How it is worked out to see the law of cosines or sines used at each step.

The laws used

The law of cosines gives a side from two sides and the angle between them, c² = a² + b² − 2ab cos γ, and an angle from three sides, cos α = (b² + c² − a²) ÷ (2bc). The law of sines says that a ÷ sin α = b ÷ sin β = c ÷ sin γ.

The three angles add up to 180°. Heron's formula gives the area from the sides alone: with s = (a + b + c) ÷ 2, the area is √(s(s − a)(s − b)(s − c)). The height on side a is 2 × area ÷ a.

c² = a² + b² − 2ab cos γ
a ÷ sin α = b ÷ sin β = c ÷ sin γ
area = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2

The five cases

SSS: three sides. The angles come from the law of cosines.
SAS: two sides and the angle between them. The third side comes from the law of cosines, then the angles.
ASA and AAS: two angles and a side. The third angle is 180° minus the other two, and the law of sines gives the sides.
SSA: two sides and an angle opposite one of them. The law of sines gives the angle opposite the other side. That angle and its supplement can both fit, which gives two triangles, or one, or none.

What the calculator does not cover

Triangles in three dimensions, triangles on a sphere, and right-triangle problems with a known hypotenuse only.
Drawings: the result is a table and the steps, not a picture.
Angles in radians: enter degrees only.

Worked example: sides 3, 4 and 5

Three sides are given, so the angles come from the law of cosines: cos α = (16 + 25 − 9) ÷ 40 = 0.8, so α = 36.87°; β = 53.13°; and γ = 90°, a right angle.

Heron's formula gives s = 6, and the area √(6 × 3 × 2 × 1) = 6. The perimeter is 12 and the heights are 4, 3 and 2.4.

	Result
Angles α, β, γ	36.87°, 53.13°, 90°
Area	6
Perimeter	12
Heights h_a, h_b, h_c	4, 3, 2.4

Frequently asked questions

Which three values can I enter?

Any three of the six: sides a, b and c, and angles α, β and γ, as long as they are not all three angles. Two sides and an angle, or two angles and a side, are the usual choices.

What is the ambiguous case?

When two sides and an angle opposite one of them are known, the side can reach the other side in two places if the given side is short. Then there are two triangles, and both are shown with their own steps.

Why does the calculator say there is no triangle?

The values cannot form a triangle. Three sides must satisfy the triangle inequality, and an SSA set has no triangle when the side opposite the given angle is too short to reach the other side.

Why are the angles 0° to 180° only?

A triangle's angles are each greater than 0° and less than 180°, and their sum is 180°, so an angle outside that range cannot be part of a triangle.

How is the area found?

The area is found by Heron's formula from the three sides, so it works whatever the known values were. The heights are then 2 × area divided by the side each is drawn to.

Are the values I enter sent to a server?

No. The triangle is solved in your browser, and nothing you type is sent to us.

Sources

Figures and text last checked on Tuesday, November 10, 2026.

Result

How it is worked out