Free Oblique Triangle Calculator

Understanding Oblique Triangles

A triangle that does not contain a 90‑degree angle is called an oblique triangle. This broad category includes acute triangles (all interior angles less than 90°), obtuse triangles (one angle greater than 90°), as well as equilateral, isosceles, and scalene shapes. Unlike right triangles, which rely on the Pythagorean theorem, an oblique triangle solver uses two core trigonometric relations: the Law of Sines and the Law of Cosines. Whether you need missing side lengths, angle measures, area, or perimeter, a dedicated Triangle Solver can turn any valid set of known values into the unknown dimensions.

The Two Pillars of Oblique Triangle Solving

Law of Sines — This proportion links each side to the sine of its opposite angle:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

When at least two angles and one side, or two sides and a non‑included angle are known, this law provides the missing values (though the SSA case requires caution).

Law of Cosines — An extension of the Pythagorean theorem that works for every triangle:

$$c^{2} = a^{2} + b^{2} - 2ab\cos C$$

Rearranging the Law of Cosines allows you to compute any angle when only the three sides are given. Together, these two formulas handle all unambiguous oblique triangle data combinations.

Solving with Three Known Sides (SSS Triangle)

Given all three side lengths (a SSS triangle), compute the semi‑perimeter first:

$$s = \frac{a + b + c}{2}$$

The area follows directly from Heron’s formula:

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

To obtain the interior angles, apply the Law of Cosines rearranged for each angle:

$$\cos A = \frac{b^{2}+c^{2}-a^{2}}{2bc},\quad \cos B = \frac{a^{2}+c^{2}-b^{2}}{2ac},\quad \cos C = \frac{a^{2}+b^{2}-c^{2}}{2ab}$$

The perimeter is simply $P = a+b+c$. A modern SSS Triangle Calculator performs these steps instantly.

Solving with Two Sides and the Included Angle (SAS Triangle)

When two side lengths and the angle between them are known (a SAS triangle), start with the Law of Cosines to find the missing side:

$$c = \sqrt{a^{2}+b^{2}-2ab\cos\gamma}\quad (\text{with }\gamma\text{ as the included angle})$$

The area is then:

$$A = \frac12 ab\sin\gamma$$

Use the Law of Sines to determine the remaining two angles. Dedicated SAS Triangle Calculator tools accept $a$, $b$, and $\gamma$ and output all other quantities.

Solving with Two Angles and a Side (ASA / AAS Triangle)

Two different data arrangements exist when you know two angles and one side.

ASA triangle — Two angles and the side that lies between them.

1. Find the third angle: $\gamma = 180° - \alpha - \beta$.
2. Apply the Law of Sines to solve for the other sides:

$$a = c\cdot\frac{\sin\alpha}{\sin\gamma},\qquad b = c\cdot\frac{\sin\beta}{\sin\gamma}$$

AAS triangle — Two consecutive angles and a side that is not between them. Again, compute the missing angle first, then use the Law of Sines to obtain the unknown sides.

For both ASA and AAS, once all three side lengths are known, area and perimeter can be computed. The corresponding ASA Triangle Calculator and AAS Triangle Calculator modes handle these cases without manual trigonometric work.

Why SSA Is Not a Valid Combination

An SSA triangle (two sides and a non‑included angle) does not define a unique shape. Depending on the given values, there may be zero, one, or two possible triangles. Because of this ambiguity, reliable triangle solvers only support the unambiguous configurations: SSS, SAS, ASA, and AAS.

Making the Most of This Free Triangle Solver Online

Using the tool is straightforward. Choose the data type that matches what you already know—SSS, SAS, ASA, or AAS—and fill in the appropriate fields. The calculator applies the Law of Sines and Law of Cosines in the correct order, returning all side lengths, all angles, the perimeter, and the area. No prior trigonometric expertise is needed, yet the step‑by‑step logic can also serve as a learning aid. Whether you are checking homework, planning a construction project, or exploring geometry for fun, an oblique triangle solver puts the power of triangle mathematics right at your fingertips.