Area of an Oblique Triangle Calculator
An oblique triangle is any triangle that has no right angle. This calculator finds its area using whichever combination of sides and angles you have: two sides and the included angle (SAS), all three sides (SSS), two angles and a side (AAS), or a base and perpendicular height. Choose a method, enter your values, and the area updates instantly along with a step-by-step explanation of the math.
What is an oblique triangle?
An oblique triangle is any triangle that contains no right angle - that is, none of its interior angles equals 90 degrees. This distinguishes it from a right triangle, which always has one 90-degree corner. Oblique triangles fall into two sub-types: acute, where every angle is less than 90 degrees, and obtuse, where one angle exceeds 90 degrees. Both types appear constantly in surveying, navigation, engineering, and architecture, where the shapes of real objects rarely line up at convenient right angles. Because there is no right angle to exploit, you cannot use the simple Pythagorean theorem directly, and instead must apply the Law of Sines, Law of Cosines, or one of the four area formulas supported by this calculator.
How to calculate the area of an oblique triangle
Four formulas cover every practical situation. If you know two sides and the angle between them (SAS), the area is half the product of those sides multiplied by the sine of the included angle: Area = (1/2) x a x b x sin(C). If you know all three sides but no angles (SSS), use Heron's formula: first compute the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s(s - a)(s - b)(s - c)). If you know two angles and a non-included side (AAS), the third angle is 180 minus the other two, and then the Law of Sines gives the missing sides so the SAS formula can be applied; the compact version is Area = a^2 x sin(B) x sin(C) / (2 x sin(A)). Finally, if the perpendicular height from the base is available, the universal formula Area = (1/2) x base x height works for any triangle regardless of its angles.
The Law of Sines and Law of Cosines in area calculations
The Law of Sines states that in any triangle the ratio of each side to the sine of its opposite angle is constant: a / sin(A) = b / sin(B) = c / sin(C). This constant ratio equals twice the circumradius of the triangle. It is the key tool for the AAS and ASA cases: given two angles you immediately know the third (because all three sum to 180 degrees), and the known side lets you scale the triangle to find the remaining sides. The Law of Cosines, c^2 = a^2 + b^2 - 2ab x cos(C), generalises the Pythagorean theorem to oblique triangles. It is used here in the SAS mode to compute the third side for the perimeter output, and it underpins Heron's formula derivation. Both laws are exact for all triangles and produce the same area regardless of which formula you start from.
Worked example: SAS oblique triangle
A triangle has sides a = 7 units and b = 10 units with an included angle C = 60 degrees. Step 1: find sin(60 degrees) = 0.8660. Step 2: Area = (1/2) x 7 x 10 x 0.8660 = 35 x 0.8660 = 30.311 square units. The third side follows from the Law of Cosines: c = sqrt(7^2 + 10^2 - 2 x 7 x 10 x cos(60)) = sqrt(49 + 100 - 70) = sqrt(79) = 8.888 units, giving a perimeter of 25.888 units. These are the default values loaded in this calculator so you can verify the output immediately.
Oblique triangle area formulas at a glance
| Known values | Formula | Notes |
|---|---|---|
| Two sides a, b and included angle C | Area = (1/2) x a x b x sin(C) | SAS - most common case |
| Three sides a, b, c | Area = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2 | Heron's formula (SSS) |
| Two angles A, B and side a (opposite A) | Area = a^2 x sin(B) x sin(C) / (2 sin(A)) | AAS - angles sum to 180 |
| Base b and perpendicular height h | Area = (1/2) x b x h | Simplest when height is known |
Choose the formula that matches what you already know.
Frequently asked questions
What is an oblique triangle?
An oblique triangle is any triangle without a 90-degree angle. It can be acute (all angles under 90 degrees) or obtuse (one angle over 90 degrees). The special right-triangle formulas do not apply directly, so you use the Law of Sines, the Law of Cosines, or one of the four area formulas above.
Which formula should I use if I only know two sides and one angle?
It depends on which angle you know. If the angle is between the two known sides (the included angle), use the SAS formula: Area = (1/2) x a x b x sin(C). If the angle is opposite one of the known sides, that is the AAS or ASA case - use the Law of Sines first to find the missing pieces, or use the AAS formula directly: Area = a^2 x sin(B) x sin(C) / (2 x sin(A)).
What is Heron's formula and when do I need it?
Heron's formula calculates the area of a triangle when you know all three side lengths but none of the angles. It works by computing the semi-perimeter s = (a + b + c) / 2 and then Area = sqrt(s(s - a)(s - b)(s - c)). It is named after Hero of Alexandria, who described it around 60 AD. The formula works for any triangle - right or oblique - and is especially useful in surveying and construction when only distances can be measured.
Can these formulas give the wrong answer?
The formulas are exact when valid inputs are supplied. Common errors include using degrees when the formula expects radians (this calculator converts for you), supplying angles that do not form a valid triangle (angles must sum to 180 and all must be positive), or using side lengths that violate the triangle inequality (each side must be shorter than the sum of the other two). The calculator returns no result rather than a wrong one in those cases.
Does the formula change for an obtuse triangle?
No. The SAS, SSS, AAS, and base-height formulas all work identically for acute and obtuse oblique triangles. When an angle is obtuse, sin(C) is still positive (sine is positive in the second quadrant), so the SAS formula gives the correct positive area. Heron's formula is purely algebraic and does not depend on angle type at all.
What is the AAS formula shortcut?
When you know two angles A and B and the side a opposite angle A, you can find the area directly as: Area = a^2 x sin(B) x sin(C) / (2 x sin(A)), where C = 180 - A - B. This is derived by using the Law of Sines to express sides b and c in terms of a, then substituting into the SAS formula. It saves you from computing the other sides separately.