Oblique Triangle Calculator

What Is an Oblique Triangle?

An oblique triangle is any triangle that does not contain a right angle ($90^\circ$). Because it lacks a perpendicular $90^\circ$ angle, basic right-triangle trigonometry (such as $\sin\theta = \text{opposite}/\text{hypotenuse}$) cannot be applied directly to its original side lengths. Instead, oblique triangles are solved using the Law of Sines and the Law of Cosines.

Oblique triangles are categorized into two primary families based on their interior angles:

Acute Oblique Triangles: All three interior angles are strictly less than $90^\circ$ ($\alpha, \beta, \gamma < 90^\circ$).
Obtuse Oblique Triangles: Exactly one interior angle is greater than $90^\circ$ (for example, $\alpha > 90^\circ$), with the remaining two angles being acute.

Fundamental Laws for Solving Oblique Triangles

1. The Law of Sines

The Law of Sines states that the ratio of the sine of an angle to the length of its opposite side is constant across all three vertices:

$$\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} = 2R$$

where $R$ is the radius of the triangle's circumscribed circle (circumradius). The Law of Sines is primarily used for ASA, AAS, and SSA (ambiguous case) scenarios.

2. The Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem to any triangle:

$$c^2 = a^2 + b^2 - 2ab\cos\gamma$$

$$a^2 = b^2 + c^2 - 2bc\cos\alpha$$

$$b^2 = a^2 + c^2 - 2ac\cos\beta$$

The Law of Cosines is indispensable when solving SAS (two sides and the included angle) and SSS (three known sides).

The 5 Solving Cases Explained

1. SSS (Side-Side-Side): When all three side lengths $a, b, c$ are known, verify the triangle inequality ($a + b > c, a + c > b, b + c > a$), then use the rearranged Law of Cosines to solve for all three angles: $$\cos\alpha = \frac{b^2 + c^2 - a^2}{2bc}$$

2. SAS (Side-Angle-Side): When two sides $a, b$ and their included angle $\gamma$ are known, calculate the third side $c = \sqrt{a^2 + b^2 - 2ab\cos\gamma}$, then find the remaining angles.

3. ASA (Angle-Side-Angle): When two angles $\alpha, \beta$ and the included side $c$ are known, find the third angle via $\gamma = 180^\circ - (\alpha + \beta)$, then apply the Law of Sines to find $a$ and $b$.

4. AAS (Angle-Angle-Side): When two angles $\alpha, \beta$ and a non-included side $a$ are known, compute $\gamma = 180^\circ - (\alpha + \beta)$ and use the Law of Sines to solve for sides $b$ and $c$.

5. SSA (The Ambiguous Case): When two sides $a, b$ and a non-included angle $\alpha$ are known, comparing side $a$ to the altitude $h = b\sin\alpha$ determines whether there are $0$, $1$, or $2$ distinct valid triangles.

Calculating Area, Inradius, and Circumradius

For any oblique triangle with semi-perimeter $s = \frac{a + b + c}{2}$:

Area via Trigonometry: $\text{Area} = \frac{1}{2}ab\sin\gamma = \frac{1}{2}bc\sin\alpha = \frac{1}{2}ac\sin\beta$
Area via Heron's Formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$
Inradius ($r$): $r = \frac{\text{Area}}{s}$
Circumradius ($R$): $R = \frac{abc}{4\cdot\text{Area}} = \frac{a}{2\sin\alpha}$

Related Triangle Tools

Explore our collection of geometry calculators:

Acute Triangle Calculator: Dedicated solver for acute oblique triangles.
3 Sides Triangle Area Calculator: Calculate area from side lengths using Heron formula.
SAS Triangle Area Calculator: Compute area and geometry for side-angle-side inputs.
ASA Triangle Calculator: Solve triangles given angle-side-angle measurements.
AAS Triangle Calculator: Solve triangles given two angles and a non-included side.
Right Triangle Calculator: Solve triangles with a 90-degree right angle.

Frequently Asked Questions

Can an oblique triangle have a 90-degree angle?

No. By definition, an oblique triangle is any triangle that has no right angle ($90^\circ$). Triangles with a $90^\circ$ angle are classified as right triangles and can be solved directly with the Pythagorean theorem.

Why is SSA called the ambiguous case?

Given two sides $a, b$ and a non-included angle $\alpha$, the geometric arrangement can produce zero solutions (if side $a$ is too short to reach the base), one unique solution (if $a$ equals the perpendicular altitude $h$ or $a \ge b$), or two distinct valid triangles (one acute and one obtuse, when $h < a < b$).

When should I use the Law of Cosines instead of the Law of Sines?

Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS). Use the Law of Sines when you know a side and its opposite angle (such as in ASA, AAS, or SSA).

What is the circumradius of an oblique triangle?

The circumradius $R$ is the radius of the unique circle that passes through all three vertices of the triangle. It can be calculated as $R = \frac{a}{2\sin\alpha} = \frac{abc}{4\cdot\text{Area}}$.

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