How to Find the Area of a Triangle From Two Sides and the Included Angle

This is the formula for the case where you can measure two sides and the angle where they meet, which happens constantly in surveying and in any layout work set out from a single corner. You do not need a height and you do not need the third side.

Where the formula comes from

It is the ordinary area formula with one substitution. Start from half base times height and take side a as the base. The height is the perpendicular from the far corner down to that base, and the far corner sits at the end of side b. Drop that perpendicular and you make a right triangle whose hypotenuse is b and whose angle at the corner is C, so the height equals b sin C.

Substitute that in and half base times height becomes half of a times b sin C. Nothing new has been introduced. The sine is simply doing the work of converting a slanted side into the perpendicular height you could not measure directly, which is exactly why the formula fails the moment you feed it an angle that is not positioned to do that job.

A worked example

Two sides of a triangular plot measure 9 meters and 12 meters, and the corner where they meet opens to 40 degrees. Multiply the sides first: 9 times 12 is 108. The sine of 40 degrees is 0.6428, so 108 times 0.6428 is 69.42. Halve that and the area is 34.71 square meters.

You can sanity check this without repeating the arithmetic. If that corner were opened all the way to 90 degrees, the sine would be 1 and the area would be half of 108, which is 54. Since 40 degrees is a much tighter corner than 90, the answer has to come out well below 54, and 34.71 does.

The angle has to be the one between the two sides

This is where the formula goes wrong for most people, and it goes wrong quietly. Every triangle has three angles, and all three are real numbers belonging to the same shape. Substituting the wrong one still produces an answer that looks perfectly reasonable.

Obtuse angles need no special treatment

Sine behaves in a way that catches people here. An angle of 140 degrees has exactly the same sine as one of 40 degrees, because sine is symmetric about 90 degrees. Feed 140 into the formula with the same sides of 9 and 12 and you get 34.71 again, identical to the answer for 40 degrees.

That is correct, not a coincidence and not a bug. The two triangles are genuinely different shapes with genuinely equal areas, because the perpendicular height b sin C works out the same in both. It does mean area alone can never tell you whether a corner is acute or obtuse, which matters if you are working backwards from an area to recover an angle. Going that direction, two answers are always possible and you need something else to decide between them.

Check the calculator mode before you trust the number

A calculator left in radians will read 40 as forty radians and return a sine of 0.745, giving an area of 40.2 instead of 34.71. That is close enough to the right answer to survive a glance and wrong enough to matter. The quickest test is sin 90, which should give exactly 1 in degree mode and 0.894 in radians.

Mistakes that catch people out

• Using an angle that is not between the two sides. The single most common error. Find the corner where your two measured sides actually meet.
• Leaving the calculator in radians. Test with sin 90 first. It should return exactly 1.
• Using cosine instead of sine. Cos 40 is 0.766 against sin 40 at 0.643, so the area comes out near 41 rather than 34.71.
• Forgetting to halve. Same trap as the ordinary area formula. Without the half you get double the area.
• Expecting the obtuse case to differ. 40 degrees and 140 degrees give the same area, and that is genuinely correct.

Practice questions

Common questions

What is the formula for the area of a triangle from two sides and an angle?

The formula is A = half of a times b times sin C, where a and b are the two side lengths and C is the angle enclosed between them.

Which angle do I use?

The one at the corner where your two known sides meet. Any other angle in the triangle gives a wrong answer even though it is a real angle of that triangle.

Why does the formula use sine and not cosine?

Sine converts the slanted second side into the perpendicular height. The height equals b sin C, and substituting that into half base times height produces the formula.

Does it work if the angle is obtuse?

Yes, with no change. An angle of 140 degrees has the same sine as one of 40 degrees, so both give the same area, and that is correct rather than an error.

What if my calculator is set to radians?

You will get a wrong but plausible looking answer. Test with sin 90, which should return exactly 1 in degree mode.

Do I need the third side?

No. Two sides and the angle between them are enough. If you only have the three sides and no angle, use Heron formula instead.

Related lessons and calculators

• Area of a Triangle: Every Formula and When to Use Each for the overview of all four methods and how to choose between them.
• How to Find the Area of a Triangle From Three Sides for when you have no angle at all and only the three lengths.
• How to Find the Height of a Triangle From the Area and Base for recovering the perpendicular height once you have the area.
• Triangle Solver which handles this two sides and included angle case along with the rest.
• Distance and Pythagoras Calculator for turning corner coordinates into the two side lengths.
• National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.