What is the difference between AAS and ASA?

Where the known side sits. In ASA it joins the two known angles, as side c joins corners A and B. In AAS it is opposite one of them, as side a is opposite A. Both are solved the same way: the third angle first, as 180 minus the other two, then the law of sines for the missing sides, so the tool handles both here and says which one you gave it. Neither can be ambiguous, so there is always exactly one answer.

How does the law of sines find the sides?

Every side divided by the sine of the angle opposite it gives the same number. Here a over sin A is 10 over 0.642788, which is 15.5572, so b is 15.5572 times sin 60, which is 13.473, and c is 15.5572 times sin 80, which is 15.3209. That is why any one side is enough once all three angles are known: it sets the scale, and the angles set the shape.

What if I only know the two angles?

Then the third angle is known and the size is not. Type just the two angles and the tool reports the third straight away, 40 and 60 leaving 80, and asks for one side before it gives any lengths or an area, because every triangle with those angles, large or small, would be an equally correct answer.

AAS and ASA Triangle Calculator

Give two angles and any one side to get the third angle from the 180 degree sum and the two missing sides from the law of sines, with area and heights. The page opens on A = 40 degrees, B = 60 degrees and side a = 10, which is angle-angle-side: C is 80 degrees, and the law of sines, a over sin A equals b over sin B equals c over sin C, gives b as 13.473 and c as 15.3209, with an area of 66.3414. Clear a and type 10 into c instead and it becomes angle-side-angle, the side between the two angles: the same three angles now give a = 6.52704 and b = 8.79385, a smaller triangle of the same shape. Two angles that already reach 180 degrees are refused with their sum shown.

Solvedone triangle

AASa 10 b 13.473 c 15.3209Degrees

a 10   b 13.473   c 15.3209
A 40°   B 60°   C 80°
area 66.3414     perimeter 38.7939
height to a 13.2683   to b 9.84808   to c 8.66025
acute, scalene

the 180 degree angle sum, then the law of sines for the two missing sides

A (°)

B (°)

C (°)

Angles in

40°

60°

80°

Area

66.3414

Two angles and a side that is not between them, so A, B and a. The third angle is what is left of 180 degrees, and the law of sines scales the triangle to the side you gave.

AAS: one triangle fits.

SSS 7, 8, 9three sides
one triangle, area 26.8328
SSA ambiguousa 7, b 9, A 40°
two triangles, both shown: B 55.7349° or 124.2651°
Impossible sides1, 2, 9
refused: These three lengths cannot meet: 1 and 2 together are shorter than 9

How the five cases work. Fill any three of the six boxes and the tool reads which case you gave it. Three sides go to the law of cosines for every angle and Heron’s formula for the area. Two sides with the angle between them get the third side from the law of cosines first. Two angles and any side get the third angle from the 180 degree sum and the missing sides from the law of sines. Two sides with an angle that is not between them are the ambiguous case: when the side opposite the given angle is shorter than the other side but longer than the height, two different triangles fit, and you get both. Three angles are refused, because they set the shape and not the size. Exact trigonometry by the law of cosines and the law of sines, with rounding clamped so a right angle does not come back as 89.999 degrees. The ambiguous side-side-angle case returns both valid triangles rather than picking one.

Common questions

What is the difference between AAS and ASA?: Where the known side sits. In ASA it joins the two known angles, as side c joins corners A and B. In AAS it is opposite one of them, as side a is opposite A. Both are solved the same way: the third angle first, as 180 minus the other two, then the law of sines for the missing sides, so the tool handles both here and says which one you gave it. Neither can be ambiguous, so there is always exactly one answer.
How does the law of sines find the sides?: Every side divided by the sine of the angle opposite it gives the same number. Here a over sin A is 10 over 0.642788, which is 15.5572, so b is 15.5572 times sin 60, which is 13.473, and c is 15.5572 times sin 80, which is 15.3209. That is why any one side is enough once all three angles are known: it sets the scale, and the angles set the shape.
What if I only know the two angles?: Then the third angle is known and the size is not. Type just the two angles and the tool reports the third straight away, 40 and 60 leaving 80, and asks for one side before it gives any lengths or an area, because every triangle with those angles, large or small, would be an equally correct answer.

Exact trigonometry by the law of cosines and the law of sines, with rounding clamped so a right angle does not come back as 89.999 degrees. The ambiguous side-side-angle case returns both valid triangles rather than picking one.