Enter the values of any three parameters of a triangle (three sides, or one side and two angles, or two sides and one angle) in the fields of the Triangle Calculator and calculate the missing parameters as well as the perimeter and the area of the triangle.
Triangle formulas
As is well known from geometry a triangle of general form is characterized by six main parameters: three side lengths (\(a = BC,\) \(b = CA,\) \(c = AB\)) and three angles (\(α = ∠A,\) \(β = ∠B,\) \(γ = ∠C\)). In the classical problem of solving plane triangles there are three parameters specified and the other three parameters are to be determined. Thus a triangle can be uniquely defined when given any of the following:
For all of these cases at least one of the side lengths must be specified. In case only the angles are specified, the side lengths cannot be determined, because any similar triangle can be a solution.
Let’s consider all of the above cases.
Given Three Sides (SSS)
When three side lengths \(a, b, c\) are specified the law of cosines can be used in order to determine the angles \(α\) and \(β\):
$$\alpha =arccos\frac { { b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 } }{ 2bc },$$
$$\beta =arccos\frac { { a }^{ 2 }+{ c }^{ 2 }-{ b }^{ 2 } }{ 2ac }.$$
Then the third angle \(γ\) can be easily found from the known angle sum property of a triangle: $$γ = 180° − α − β.$$
Given Two Sides and the Included Angle (SAS)
When the lengths of sides \(a, b\) and the angle \(γ\) between them are given the third side c can be determined from the law of cosines:
$$c = {\sqrt { { a }^{ 2 }+{ b }^{ 2 }-2ab\cdot cos\gamma } }.$$
Then the law of cosines can be used to determine the second angle \(α\):
$$\alpha ={arccos\frac { { b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 } }{ 2bc }}.$$
And the third angle \(β\) can be finally found from \(β = 180° − α − γ\).
Given Two Sides and Non-included Angle (SSA)
This is the most complicated option because there are cases when a solution does not exist at all or there are more than one solution.
A unique solution is guaranteed to exist only if the side length adjacent to the given angle is shorter than the other side length. Let the two sides \(b, c\) and the angle \(β\) are given. The value of the angle \(γ\) can be determined from the law of sines:
$$sin\gamma =\frac { c }{ b } sin\beta. $$
Let’s denote the right side of the above equation as \(D=({ c }/{ b)\cdot }sin\beta \). Then we have four possible cases:
- \(D > 1\). Such a triangle does not exist because the side \(b\) does not reach the line \(BC\). For the same reason a solution does not exist in case the angle \(β ≥ 90°\) and \(b ≤ c\).
- \(D = 1\). There is a unique solution: \(γ = 90°\), i.e., the triangle is right-angled.
- \(D < 1\). There are two possible alternatives.
- If \(b ≥ c\) then \(β ≥ γ\) i.e. the larger side corresponds to a larger angle. Since there is no triangle with two obtuse angles, \(γ\) is an acute angle and the solution \({\gamma =arcsinD\quad}\) is unique.
- If \(b < c\), the angle \(γ\) may be either acute: \({\gamma =arcsinD\quad}\) or obtuse: \(γ′ = 180° – γ\). So in this case we have two solutions.
Then the third angle \(α = 180° − β − γ\).
And, finally, the third side length can be found from the law of sines:
$$a=b\cdot \frac { sin\alpha }{ cos\beta } $$
or
$$a={c\cdot cos{\beta} \pm \sqrt { { b }^{ 2 }-{ c }^{ 2 }\cdot \sin ^{ 2 }{ \beta } } }.$$
Given A Side and Two Adjacent Angles (ASA)
Let the given parameters are the side \(c\) and the angles \(α, β\). The third angle \(γ = 180° − α − β\). Two unknown side lengths can be determined from the law of sines:
$$a=c\cdot \frac { sin\alpha }{ sin\gamma } ;\quad b=c\cdot \frac { sin\beta }{ sin\gamma } ,$$
or
$$a=c\cdot \frac { sin\alpha }{ sin\alpha \cdot cos\beta +sin\beta \cdot cos\alpha } ,$$
$$b=c\cdot \frac { sin\beta }{ sin\alpha \cdot cos\beta +sin\beta \cdot cos\alpha }.$$
Given A Side, One Adjacent Angle and the Opposite Angle (AAS)
For solving an AAS triangle we can use the same procedure as for an ASA triangle. First of all, we can determine the third angle value by using the angle sum property of a triangle, and then determine the other two side lengths using the law of sines.
Related calculators
Check out our other geometry calculators such as Circle Calculator or Sphere Calculator.