What the Oblique Triangle Area Calculator does and when to use it
An oblique triangle is any triangle without a right angle, so the familiar half base times height shortcut is not directly available from the given data. This calculator handles the common case where you know two sides and the angle between them (side-angle-side). It returns the area, the length of the missing third side, the perimeter, and the height measured onto the first side.
It is useful in surveying, roofing, land plots, sailing and any drawing where you can measure two edges and the angle they form but not the vertical height. Enter the two lengths in any single unit, and the angle in degrees; the outputs use the same unit, squared for area.
Formula and method
The area of a triangle with sides a and b and included angle C is A = ½ × a × b × sin C. The third side follows from the law of cosines, c² = a² + b² − 2ab cos C. The perimeter is a + b + c, and the height onto side a is h = b × sin C, which is the same expression that appears in the area formula.
The angle must be strictly between 0 and 180 degrees. Angles above 90 degrees are allowed and describe obtuse triangles.
- a, b the two known side lengths.
- C the angle between sides a and b, in degrees.
- c the side opposite angle C.
- h the perpendicular height from the far vertex onto side a.
- A the triangle's area.
Worked example
Take sides a = 7 and b = 10 with an included angle C = 40 degrees.
- sin 40° = 0.642788 and cos 40° = 0.766044.
- Area = 0.5 × 7 × 10 × 0.642788 = 22.498.
- c² = 49 + 100 − 140 × 0.766044 = 41.754, so c = 6.462.
- Perimeter = 7 + 10 + 6.462 = 23.462.
- Height on side a = 10 × 0.642788 = 6.428.
These match the calculator's results. You can cross-check with Heron's formula: the semi-perimeter is 11.731, and the square root of 11.731 × 4.731 × 1.731 × 5.269 is about 22.50. If the angle were 140 degrees instead, the area would stay 22.498 (sine of 140° equals sine of 40°) but the third side would grow to 16.008.
Common mistakes and how to interpret the result
- Using the wrong angle. The formula needs the angle between the two known sides. An angle at a different corner requires the law of sines first.
- Entering radians. The field expects degrees; typing 0.7 for about 40 degrees will produce a sliver-thin triangle.
- Mixing units for the sides. Both sides must share one unit; convert feet and inches before entering, and remember the area is in square units.
- Assuming the height lies inside the triangle. For obtuse angles the perpendicular foot falls outside the base, though the numeric height and area are still correct.
Related calculators
- Area of Triangle (SAS) — the same idea with a focused layout.
- Law of Cosines Calculator — solve sides and angles with cosines.
- Triangle Area from Three Sides — use Heron's formula.
- Obtuse Triangle Area — area when one angle exceeds 90 degrees.