Area of an Oblique Triangle Calculator
An oblique triangle is any triangle without a right angle. Enter two sides and the angle between them; the calculator returns the area via ½·a·b·sin(C), plus the missing side and angles from the laws of cosines and sines.
Area
17.3205 square units
- Third side c (law of cosines)
- 7
- Angle A
- 81.79°
- Angle B
- 38.21°
- Angle C (included)
- 60°
- Perimeter
- 20
- Classification
- acute oblique
Oblique triangle area uses the SAS formula: area = ½·a·b·sin(C), where C is the included angle. The third side comes from the law of cosines, c² = a² + b² − 2ab·cos(C), and the other two angles from the law of sines.
How to use this calculator
Enter the two known sides a and b in the same length unit (cm, m, in, ft — anything, as long as both use the same one), then enter the included angle C — the angle between sides a and b — in degrees. The headline result is the area in square units of whatever length unit you used. The breakdown also gives you the third side c via the law of cosines, the other two angles via the law of sines, the perimeter, and a classification (acute oblique vs obtuse oblique) so you know which family of oblique triangle you have.
How the calculation works
An oblique triangle is any triangle that does not contain a 90° angle — i.e. every triangle except right triangles. The standard area formula for SAS (side-angle-side) data is area = ½·a·b·sin(C), where C is the angle included between sides a and b. This is exact for any triangle, not just oblique ones; for an oblique triangle, sin(C) is strictly between 0 and 1 (since C ≠ 0°, 90°, or 180°) when C is acute, equal to sin(180° − C) when C is obtuse. The third side comes from the law of cosines: c² = a² + b² − 2ab·cos(C). Once c is known, the law of sines (a/sin(A) = b/sin(B) = c/sin(C)) gives the remaining two angles. The triangle is acute oblique if all three angles are below 90°, and obtuse oblique if one angle exceeds 90°.
Worked example
Take a = 8, b = 5, C = 60°. Then sin(60°) = √3/2 ≈ 0.8660, so area = ½ · 8 · 5 · 0.8660 ≈ 17.32 square units. The third side from the law of cosines is c² = 64 + 25 − 2·8·5·cos(60°) = 89 − 40 = 49, so c = 7. The remaining angles via the law of sines: sin(A) = a·sin(C)/c = 8·0.8660/7 ≈ 0.9897, giving A ≈ 81.79°; then B = 180° − 60° − 81.79° ≈ 38.21°. Every angle is below 90°, so the triangle is acute oblique.
Frequently asked questions
What is an oblique triangle?
An oblique triangle is any triangle that does not contain a right angle. Triangles come in two broad families: right triangles, which have exactly one 90° angle, and oblique triangles, which have none. Oblique triangles split further into acute oblique (every angle below 90°) and obtuse oblique (one angle above 90°). The "oblique" label exists because the techniques for solving these triangles — the law of sines and law of cosines — differ from the simpler Pythagorean theorem that works only for right triangles.
What is the area formula for an oblique triangle?
The most common formula is the SAS (side-angle-side) area, area = ½·a·b·sin(C), where a and b are two sides and C is the angle between them. This formula works for every triangle, not just oblique ones — but it is especially useful for oblique triangles because there is no convenient base-and-height to read off. If instead you know all three sides, use Heron's formula: area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. Both formulas give exactly the same answer when applied to the same triangle.
How does the included angle differ from the other two angles?
The included angle is the angle formed where two named sides meet. If your two sides are a and b, the included angle is the one wedged between them — by convention labelled C, because by the standard naming convention each angle takes the name of the side opposite to it (so angle C sits across from side c). The non-included angles, A and B, are opposite the named sides a and b. Only the included angle works directly in the ½·a·b·sin(C) formula; using a non-included angle would give the wrong area.
Why does the ½·a·b·sin(C) formula work?
Drop a perpendicular from the vertex opposite side a (or equivalently side b) onto the base. The height h of that perpendicular equals b·sin(C), because in the right triangle formed by the perpendicular, b is the hypotenuse and h is the side opposite the angle C. The triangle area is then ½ · base · height = ½ · a · b·sin(C) = ½·a·b·sin(C). The formula reduces to the familiar ½·base·height when C = 90° (because sin(90°) = 1), so it generalises that simpler case to all triangles.
What happens if the included angle is 90°?
The triangle is no longer oblique — it is a right triangle, with the right angle at C. The formula area = ½·a·b·sin(C) still applies and simplifies to ½·a·b (since sin(90°) = 1), which is just ½·base·height with the two legs of the right triangle as base and height. The calculator will return a valid answer, but flag the classification differently — and you can also use the dedicated right triangle calculator for that case.
Can I use this calculator if I only know the three sides?
Not directly — this calculator expects two sides and the included angle. If you have three sides, use the Heron's formula calculator or the general triangle calculator, both linked below. Alternatively, recover the included angle from the law of cosines first: cos(C) = (a² + b² − c²) / (2ab), then C = arccos of that value, and feed those a, b, C into this calculator.