Even though triangles have only three sides and three angles, the data combinations are quite a lot: discover the many ways to solve an oblique triangle with our oblique triangle calculator.

What is an oblique triangle?

An oblique triangle is any triangle without any particular constraint on the values of sides and angles, apart from the fundamental one: an oblique triangle can't have a right angle.

🙋 If you are looking for the mathematics of right triangles, head to our other calculators:

The definition of an oblique triangle includes all of the other types of triangles: obtuse, acute, equilateral, isosceles, and scalene.

There are no specific rules to solve an oblique triangle: we need to rely mostly on the two fundamental tools of the field:

Let's see the possible ways we have to solve and calculate an oblique triangle!

Solving an oblique triangle knowing three sides

This oblique triangle solver uses three sides to calculate a unique oblique triangle. Given aa, bb, and cc, what do we do?

To calculate the perimeter and the area, we can skip every angle calculation. For the perimeter, we sum the given sides:

P=a+b+cP = a+b+c

For the area, we can apply the Heron's formula:

A ⁣= ⁣P2 ⁣ ⁣( ⁣P2 ⁣ ⁣a ⁣) ⁣ ⁣( ⁣P2 ⁣ ⁣b ⁣) ⁣ ⁣( ⁣P2 ⁣ ⁣b ⁣)A \!=\! \sqrt{\frac{P}{2}\!\!\left(\!\frac{P}{2}\!-\!a\!\right)\!\!\left(\!\frac{P}{2}\!-\!b\!\right)\!\!\left(\!\frac{P}{2}\!-\!b\!\right)}

Technically we can compute the angles of the triangle using the angle version of the law of cosines. Here is the formula for the third angle. You can derive the others on your own; they are fairly similar!

cos(γ)=a2+b2c22abcos(\gamma)= \frac{a^2+b^2-c^2}{2\cdot a\cdot b}

Calculate an oblique triangle with two adjacent sides and the angle between them

This combination of data is usually denoted by the acronym SAS. You need to know the angle between the sides because it's the only one univocally defining a triangle. Other angles (as in the SSA combination) would allow for more than a possible shape. We will show you the calculations for the combination aa, bb, and γ\gamma.

We can calculate the area right away:

A=12absinγA = \frac{1}{2}\cdot a \cdot b \cdot \sin{\gamma}

And apply the law of cosines to find the third side:

c=a2+b22absinγc = \sqrt{a^2+b^2 - 2\cdot a\cdot b \cdot \sin{\gamma}}

The other quantities follow: use the law of sines to find the other angles, or simply ask our oblique triangle calculator to do it for you!

Calculations to solve an oblique triangle knowing two angles and a side

There are two possible ways to solve an oblique triangle knowing two sides and an angle:

  • ASA, where we know the two angles lying on a side (of which we know the length); or
  • AAS, where we know two consecutive angles and one of the two sides not comprised between them.

ASA oblique triangle solver

Here we know any combination of two angles and the side between them, for example, β\beta, γ\gamma, and aa.

In this situation, firstly compute the value of the third angle:

α=180°βγ\alpha = 180\degree - \beta -\gamma

Then apply the law of the sines to find the other two sides. Use the following equalities:

asinα=bsinβ=csinγ\frac{a}{\sin{\alpha}}=\frac{b}{\sin{\beta}}=\frac{c}{\sin{\gamma}}

And derive:

b=sinβasinαc=sinγasinα\begin{align*} b &= sin{\beta}\cdot\frac{a}{\sin{\alpha}}\\ \\ c &= sin{\gamma}\cdot\frac{a}{\sin{\alpha}} \end{align*}

Proceed to calculate the perimeter and the area with your preferred formulas!

AAS oblique triangle solver

Assume you know the angles β\beta and γ\gamma, and the side bb.

Start by calculating the last angle:

α=180°βγ\alpha = 180\degree - \beta -\gamma

Then, use again the law of the sines, but with regards to the other angle:

a=sinαbsinβc=sinγbsinβ\begin{align*} a &= sin{\alpha}\cdot\frac{b}{\sin{\beta}}\\ \\ c &= sin{\gamma}\cdot\frac{b}{\sin{\beta}} \end{align*}

How to use our oblique triangle calculator

To use our oblique triangle calculator, select the type of data you know: three sides, two sides and an angle, and so on. We will change the visible variables to fit the problem.

Fill the available fields, and find the results!

Other triangle calculators

Here at Omni Calculator, we studied triangles from every possible side — wait, this doesn't sound so much. We made many tools related to this fundamental shape of geometry: discover triangles with our calculators:

  • Triangle area;
  • AAA triangle;
  • Midsegment of a triangle;
  • Acute triangle;
  • Circumcenter of a triangle;
  • Triangle congruence;
  • Obtuse triangle;
  • Base of a triangle;
  • AAS triangle;
  • SAS triangle;
  • SSS triangle; and
  • ASA triangle.

FAQ

What's the area of a triangle with sides a = 4 cm, b = 5 cm, and angle between them γ = 40°?

The area is A = 6.428 cm². This is an SAS triangle, which means that you provided two sides and the angle between them. We can apply the formula for the area of an SAS triangle:
A = 0.5 × a × b × sin(γ) = 0.5 × 4 × 5 × 0.6428 = 6.428 cm²

Can you solve an SSA triangle?

No. An SSA triangle is not unambiguously defined by the given combination: in fact, you can find two triangles which respect the combination. Imagine the given side not adjacent to the given angle: it can be in two positions, one forming an acute angle with the other side, the other forming an obtuse angle.

What are the possible combinations of sides and angles to solve an oblique triangle?

You can solve an oblique triangle if you know:

  • Three sides (SSS triangle);
  • Two sides and the angle between them (SAS triangle);
  • Two angles and the side between them (ASA triangle); or
  • Two angles and one of the two sides not lying between them (AAS triangle).

How do I solve an AAS triangle?

To solve an AAS triangle, assuming you know α, β, and b, follow these steps:

  1. Calculate the third angle using the sum of the interior angles in a triangle: γ = 180° - α - β.
  2. Calculate the other sides using the law of sines:
    • c = sin(γ) × b/sin(ß)
    • a = sin(α) × b/sin(ß)

3 Calculate perimeter and area.

How do I solve an ASA triangle?

To solve an ASA triangle, assuming you know α, β, and c, follow these steps:

  1. Calculate the third angle using the sum of the interior angles in a triangle: γ = 180° - α - β.
  2. Calculate the other sides using the law of sines:
    • b = sin(ß) × c/sin(γ)
    • a = sin(α) × c/sin(γ)

3 Calculate perimeter and area.

Davide Borchia
Given
three sides (SSS)
triangle_image_1
SSS inputs
a
cm
b
cm
c
cm
SSS results
α
deg
ß
deg
γ
deg
Area
cm²
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