Oblique Triangle Calculator

Solve non-right triangles using the Law of Sines and Law of Cosines with step-by-step solutions.

Select Mode & Enter Known Values

SSS: Three sides known. Uses Law of Cosines to find all angles.

Result

Area
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square units
Side a--
Side b--
Side c--
Angle A--
Angle B--
Angle C--
Perimeter--
Area--

Step-by-Step Solution

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Understanding Oblique Triangles

An oblique triangle is any triangle that does not contain a right angle (90 degrees). All angles are either acute (less than 90 degrees) or one angle is obtuse (greater than 90 degrees). To solve oblique triangles, we use the Law of Sines and the Law of Cosines, since the basic trigonometric ratios (SOH-CAH-TOA) only apply directly to right triangles.

Laws Used to Solve Oblique Triangles

Law of Cosines

Relates sides and angles. Used for SSS and SAS cases.

c² = a² + b² - 2ab cos(C)

Law of Sines

Relates sides and opposite angles. Used for ASA and AAS.

a/sin(A) = b/sin(B) = c/sin(C)

Area Formula

Area using two sides and included angle.

Area = (1/2)ab sin(C)

Solving Methods by Case

SSS (Three Sides Known)

When all three sides are known, use the Law of Cosines to find each angle. First find one angle using cos(A) = (b² + c² - a²) / (2bc), then find the remaining angles similarly or use the fact that all angles sum to 180 degrees.

SAS (Two Sides and Included Angle)

Use the Law of Cosines to find the unknown side, then use the Law of Sines or Law of Cosines again to find the remaining angles.

ASA (Two Angles and Included Side)

Find the third angle (180 - A - B = C), then use the Law of Sines to find the unknown sides.

AAS (Two Angles and Non-Included Side)

Find the third angle, then use the Law of Sines to find the remaining sides.

Triangle Validity

  • The sum of all angles must equal 180 degrees.
  • The triangle inequality: any side must be less than the sum of the other two sides.
  • All sides and angles must be positive.
  • In the SSA (ambiguous) case, there may be zero, one, or two valid triangles.

Applications

Oblique triangle calculations are essential in surveying, navigation, astronomy, architecture, and engineering. They allow us to determine distances and angles that cannot be measured directly, such as the height of a mountain, the distance across a lake, or the position of a ship at sea.