Area of an Oblique Triangle: the ½·a·b·sin(C) formula, with worked examples

What an oblique triangle is — and isn’t

An oblique triangle is any triangle that does not have a 90° angle. That is the entire definition. The triangle family splits cleanly into two halves: right triangles (exactly one 90° angle) and oblique triangles (no 90° angle), and every triangle on the plane falls into exactly one of those two camps. Oblique triangles split further into acute oblique (every angle below 90°) and obtuse oblique (one angle above 90°), but for the purpose of finding the area, that sub-classification does not change anything — the same formula handles both. The area of an oblique triangle calculator takes two side lengths and the angle between them and returns the area without you needing to identify which sub-class you are in.

The reason the word "oblique" exists at all is that the Pythagorean theorem and the simple ½·base·height routine you learn first only apply painlessly to right triangles. Once you take away the right angle, you have to reach for the law of cosines and the law of sines, and the area formula has to do its own work rather than reading the height off the diagram. None of that is harder — it is just a different toolkit, and the SAS area formula ½·a·b·sin(C) is the single most useful tool in it.

The SAS area formula in one line

If you know two sides of a triangle, call them a and b, and the angle between them, call it C, the area is:

area = ½ · a · b · sin(C)

That is it. There is no separate formula for "oblique" triangles — this one works for every triangle, right or oblique, acute or obtuse. The reason it deserves its own name is the data pattern it consumes: two sides and the included angle, the angle wedged directly between the two sides you named. Geometers call this the SAS (side-angle-side) configuration. The oblique triangle area calculator is built around exactly this configuration: type a, b, C, get the area.

A common point of confusion is the word "included". If you name two sides a and b, the included angle is the one where a and b meet at a vertex — labelled C by the standard convention (each angle takes the name of the side opposite it, so angle C sits across from the unnamed third side c). The two non-included angles, A and B, sit opposite a and b respectively. Plugging a non-included angle into the formula gives nonsense; only the included one carries the geometry the formula expects.

Why ½·a·b·sin(C) is true

Start with the formula you already trust: area equals one half times base times height. Pick side a as the base. The height of the triangle is the perpendicular distance from the vertex opposite a down to the line containing a. Now look at the right triangle formed by that perpendicular: its hypotenuse is side b, and the angle at the foot of b is the included angle C. Trigonometry on the right triangle gives height = b·sin(C). Substitute back: area = ½·base·height = ½·a·(b·sin(C)) = ½·a·b·sin(C).

This derivation does not assume the triangle is acute or obtuse — only that you can talk about "the perpendicular from the opposite vertex". For an obtuse triangle where the included angle is greater than 90°, the foot of the perpendicular falls outside the segment a, but the construction still works and sin(C) is still the right scalar to multiply b by, because sin is symmetric around 90° (sin(180° − C) = sin(C)). The formula reduces to the ordinary ½·base·height in the right-triangle special case (C = 90°, sin(C) = 1), so it generalises the simpler rule rather than replacing it.

Worked example: a = 8, b = 5, C = 60°

Take the default values the area of an oblique triangle calculator starts with: a = 8, b = 5, C = 60°.

1. Sanity check the included angle. 60° is strictly between 0° and 180°, neither degenerate nor a right angle. A genuine oblique triangle exists for these inputs.
2. Look up sin(60°). sin(60°) = √3 / 2 ≈ 0.86603.
3. Apply the formula. area = ½ · 8 · 5 · 0.86603 = 20 · 0.86603 ≈ 17.32 square units. The exact value is 10√3.
4. Recover the third side via the law of cosines. c² = a² + b² − 2ab·cos(C) = 64 + 25 − 2·8·5·cos(60°) = 89 − 80·(0.5) = 89 − 40 = 49, so c = 7. The triangle turns out to be a clean integer triangle with sides 5, 7, 8.
5. Recover the remaining angles via the law of sines. sin(A)/a = sin(C)/c gives sin(A) = a·sin(C)/c = 8·0.86603/7 ≈ 0.98974, so A = arcsin(0.98974) ≈ 81.79°. Then B = 180° − A − C ≈ 38.21°. Every angle sits below 90°, so the triangle is acute oblique.
6. Cross-check with Heron’s formula. The semi-perimeter is s = (5 + 7 + 8) / 2 = 10. Heron gives area = √(10·5·3·2) = √300 = 10√3 ≈ 17.32. Identical to the SAS answer, as it must be.

Punch the same a, b, C into the calculator and you should see exactly these results: area ≈ 17.32 square units, third side c = 7, angle A ≈ 81.79°, angle B ≈ 38.21°, classification "acute oblique". If you instead know all three sides, the general triangle calculator takes SSS data and gives you the same answer the other way round.

Acute oblique versus obtuse oblique

The SAS formula does not care whether the triangle is acute oblique or obtuse oblique — it returns the correct area for both. But the two sub-classes look different and have different practical traps.

Acute oblique

All three angles are below 90°. The included angle C is below 90°, which means cos(C) is positive, which by the law of cosines means c² < a² + b² — the side opposite C is shorter than the right-triangle prediction. The 5-7-8 triangle above is acute oblique, as is any equilateral triangle (every angle exactly 60°). The altitudes all land inside the triangle and the circumcentre sits inside it too.

Obtuse oblique

Exactly one angle is greater than 90°. That obtuse angle is not always the included angle C — it could be A or B, in which case C itself is acute and the calculator's "classification" reflects which vertex hosts the obtuse angle. With a = 8, b = 5, C = 120°, for instance, sin(120°) = sin(60°) ≈ 0.86603, so the area is still 17.32 — but the third side jumps to c = √(64 + 25 + 40) = √129 ≈ 11.36, the triangle is now obtuse at C, and the other two angles shrink. The same formula handles it cleanly because sin is symmetric: sin(C) = sin(180° − C), so swapping an acute included angle for its obtuse supplement keeps the area identical even though the triangle's shape changes.

The other area formulas, and when to use them

Heron’s formula — when you know all three sides

If you have a, b, c rather than two sides and the included angle, use Heron’s formula. With semi-perimeter s = (a + b + c) / 2, area = √(s(s − a)(s − b)(s − c)). Heron handles every triangle — right, acute oblique, obtuse oblique — without any reference to angles at all. The result is identical to what ½·a·b·sin(C) gives once you compute C from the law of cosines, but Heron is more direct when sides are the natural inputs. For SSS data, point the general triangle calculator at it instead — it runs Heron under the hood and also returns the angles.

½·base·height — when you have a height

If you already know a base and a perpendicular height (say, from a surveyed plot drawn on graph paper), there is no need to find an included angle at all. Area = ½·base·height is the universal one-liner. The trick is that "height" must mean the perpendicular distance to the line containing the base, not the slant distance to the opposite vertex along another side — in an obtuse triangle the perpendicular foot can sit outside the base segment, and that is still the correct height to use.

Coordinate (shoelace) formula — when you have vertex coordinates

Given three points (x₁, y₁), (x₂, y₂), (x₃, y₃), the area is ½ · |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. This is a special case of the shoelace formula and is the natural choice when the triangle comes from a CAD file, a GIS layer, or any other source that hands you vertices rather than side lengths and angles.

Two angles and a side — convert first

If you have AAS or ASA data — two angles and any side — use the law of sines to find the other two sides, then apply ½·a·b·sin(C). Equivalently, once you have three sides, hand the problem to Heron. There is no clean one-step formula that consumes AAS or ASA data directly, so converting is the right move.

Common mistakes

• Using a non-included angle. If your two named sides are a and b, the angle that goes into ½·a·b·sin(C) is the one between them, not one of the other two. Plugging in angle A or B by mistake gives an answer off by a multiplicative factor of sin(other angle) / sin(true included angle), and the only way to spot it is to cross-check against Heron. The area of an oblique triangle calculator labels the angle field "included angle C" to head this off, but you still have to know which of your two sides counts as a and b.
• Putting the angle in radians. The calculator expects degrees. If you type 1.047 (radians for 60°), it will treat 1.047 as 1.047° — about 0.018 radians — and return an area roughly 50 times too small. Convert radians to degrees by multiplying by 180/π first, or just type the degree value.
• Mixing units between a and b. If a is in metres and b is in centimetres, the area comes out in metre·centimetres, which is not a meaningful unit. Convert both sides to the same length unit before typing them in — the area will then be in the square of that unit.
• Forgetting the ½. a·b·sin(C) is twice the triangle area; it is also the area of the parallelogram spanned by a and b, so the formula is not nonsense without the half — it just answers a different question. If your answer looks twice as big as it should, the half is the first thing to check.
• Inputs that don’t make a triangle. An angle of 0° or 180° degenerates the triangle into a line segment, area zero. The calculator clamps the angle to the open interval (0°, 180°) and refuses to classify inputs outside it.

Where SAS area calculations actually come up

The classic real-world setting is land surveying: you stand at a vertex, measure two sight lines to landmarks (sides a and b) and the angle between them (included angle C), and you want the area of the triangle they sweep out. SAS is the cheapest data to collect — one angle measurement, two distance measurements — and ½·a·b·sin(C) turns it directly into area. Trigonometric surveying is built on chains of these triangles.

It also shows up anywhere two vectors meet at a known angle. The magnitude of the cross product of two 2D vectors of lengths a and b at angle C is a·b·sin(C), exactly twice the triangle area. So computing the triangle area from SAS data is the same operation as computing a signed parallelogram area from two edge vectors — useful in computer graphics, rigid-body physics, and any geometric algorithm that decomposes polygons into triangles. The general triangle calculator wraps the same arithmetic in a friendlier interface; the oblique-specific calculator is the narrow version for when SAS is all you have.

When the SAS calculation isn’t enough

Two situations push you out of "type three numbers, read off the area" territory. The first is uncertainty: if your sides and angle came from a measurement with non-trivial error, the area inherits that error and you should propagate it explicitly — small errors in C, in particular, can produce surprisingly large errors in area when C sits near 0° or 180°, because sin is flat there. The second is non-planar triangles: a triangle drawn on the surface of a sphere (relevant for navigation or global surveying) does not obey the flat-plane ½·a·b·sin(C) at all, because its three angles sum to more than 180°. For curved-surface triangles you need spherical or hyperbolic trigonometry, which is a different toolkit altogether and out of scope for the flat-plane calculator.

Within the flat-plane case, though, ½·a·b·sin(C) is the end of the story. The area of an oblique triangle calculator runs the SAS formula, the law of cosines for c, and the law of sines for the missing angles in one go, so you only have to type the three numbers you already know.

Related calculators and where to go next

For triangles where the data you have isn’t SAS, the general triangle calculator handles SSS, SAS, ASA, and AAS configurations in one interface and is the right starting point. If you specifically want to test whether your triangle is acute oblique versus obtuse oblique from the sides alone, the acute triangle calculator does the sharpened Pythagorean test for you. For right triangles — the case the oblique formula reduces to when C = 90° — the Pythagorean theorem calculator is the direct tool. And for area in other planar shapes, the ellipse area calculator and the surface area calculator cover ellipses and 3D solids respectively.

Frequently asked questions

See the FAQ section below the calculator on the area of an oblique triangle calculator page for direct answers to the questions that come up most often — what counts as oblique, why ½·a·b·sin(C) works, the difference between included and non-included angles, how to handle the case when only the three sides are known, and what happens at the right-angle boundary. The same FAQ items are marked up with FAQPage schema so search engines can surface them directly.