Non-Right Triangle — Definition, Formula & Examples

A non-right triangle, also called an oblique triangle, is any triangle that has no 90° angle. It is either acute, with all three angles smaller than 90°, or obtuse, with one angle larger than 90°.

A triangle $ABC$ is oblique if none of its interior angles measures $90^\circ$ . By the converse of the Pythagorean theorem, this holds exactly when no labeling of its side lengths satisfies $a^2 + b^2 = c^2$ . With $c$ as the longest side, the triangle is acute if $c^2 < a^2 + b^2$ and obtuse if $c^2 > a^2 + b^2$ .

How It Works

Right-triangle ratios like

\sin\theta = \tfrac{\text{opp}}{\text{hyp}}

need a hypotenuse, and an oblique triangle does not have one. Use the law of cosines when you know two sides and the angle between them (SAS) or all three sides (SSS). Use the law of sines,

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

, when you know two angles and a side (AAS or ASA). With two sides and an angle that is not between them (SSA), the given information may fit zero, one, or two triangles, so check every case.

Worked Example

Problem: A triangle has sides

a = 5

and

b = 8

, and the angle between them is

C = 60^\circ

. Find the third side

c

Step 1: You know two sides and the included angle (SAS), so substitute into the law of cosines.

c^2 = 5^2 + 8^2 - 2(5)(8)\cos 60^\circ

Step 2: Use

\cos 60^\circ = \tfrac{1}{2}

and simplify.

c^2 = 25 + 64 - 80 \cdot \tfrac{1}{2} = 89 - 40 = 49

Step 3: Take the positive square root, because a side length must be positive.

c = \sqrt{49} = 7

Answer:

c = 7

. The triangle is oblique. Its longest side is 8, and

8^2 = 64 < 5^2 + 7^2 = 74

, so it is acute.

Why It Matters

Surveyors and navigators rarely get a convenient right angle. A ship's course plotted from two compass bearings, or the distance across a lake measured from two shoreline points, forms an oblique triangle. Precalculus and physics courses then use the same methods to find the magnitude of a resultant force or velocity vector.

Common Mistakes

Mistake: Applying SOH-CAH-TOA directly to an oblique triangle, for example writing

\cos C = \tfrac{a}{c}

Correction: Those ratios are defined only for the acute angles of a right triangle, where one side is the hypotenuse. In an oblique triangle, use the law of sines or the law of cosines. Another option is to draw an altitude, which splits the triangle into two right triangles.

Related Terms

Golden Triangle — the broader concept this term builds on
Equiangular Triangle — the broader concept this term builds on
Acute Triangle — the broader concept this term builds on
Triangle — the broader concept this term builds on
Law of Cosines — Solves oblique triangles from SAS or SSS
Cosine (cos θ) — Function used inside the law of cosines
Circle Trig Definitions — Define trig values for obtuse angles
Bearing in Math — Navigation problems that produce oblique triangles
Cosecant (csc θ) — Reciprocal of sine, which appears in law of sines