How can I tell how many triangles an SSA problem has?

Compare the side opposite the given angle with two lengths: the height, which is the other side times the sine of the angle, and the other side itself. With a = 8, b = 10 and A = 30 degrees the height is 5. If a is shorter than 5 no triangle closes. If a is exactly 5 there is one, with a right angle at B. If a is between 5 and 10 there are two. If a is 10 or more there is one. When the given angle is 90 degrees or more, a has to be longer than b for any triangle at all, and then there is exactly one. The tool works this out for you and the readout says one triangle or two triangles.

Why are the two answers so different?

Because the law of sines gives the sine of angle B, and two angles below 180 degrees share every sine: B and 180 minus B. With these numbers that is 38.6822 and 141.3178 degrees. Both leave room for angle C, so both close, and they make a long triangle with c at 14.9053 and a short one with c at 2.41526. A textbook question usually asks for both; if a drawing or a measurement tells you which corner is obtuse, that picks one of them.

Which boxes count as side-side-angle?

Any two sides with an angle that sits opposite one of them rather than between them: a and b with A or B, b and c with B or C, a and c with A or C. Type the angle between the two sides instead, C for a and b, and it is the SAS case, which always has exactly one answer; the tool reads which case you gave it and says so above the result.

SSA Triangle Calculator

Give two sides and an angle that is not between them and see whether no triangle, one or two fit, with both solved and drawn when two do. The page opens on a = 8, b = 10 and A = 30 degrees. The height from corner C down to the base is b times the sine of A, which is 5, and side a at 8 is longer than that but shorter than b, so it meets the base in two places. The first triangle has B at 38.6822 degrees and c at 14.9053; the second has B at 141.3178 degrees and c at 2.41526. Both are drawn, side by side on a wide screen and one above the other on a phone, with their angles, area, perimeter and heights, labelled first and second. Change a to 4 and the page says no triangle closes and how long a must be; change it to 12 and only one triangle fits. The law of sines gives the second angle, the 180 degree sum the third, and the law of sines again the last side.

first triangle

second triangle

Solvedtwo triangles

SSAa 8 b 10 c 14.9053Degrees

first triangle
a 8   b 10   c 14.9053
A 30°   B 38.6822°   C 111.3178°
area 37.2631     perimeter 32.9053
height to a 9.31578   to b 7.45263   to c 5
obtuse, scalene

second triangle
a 8   b 10   c 2.41526
A 30°   B 141.3178°   C 8.6822°
area 6.03814     perimeter 20.4153
height to a 1.50954   to b 1.20763   to c 5
obtuse, scalene

law of sines for the second angle, the 180 degree sum, then the law of sines for the last side

A (°)

B (°)

C (°)

Angles in

30°

38.6822°

111.3178°

Area

37.2631

Two sides and an angle that is not between them, so a, b and A. This is 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 both are shown.

SSA: two triangles fit.

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

How can I tell how many triangles an SSA problem has?: Compare the side opposite the given angle with two lengths: the height, which is the other side times the sine of the angle, and the other side itself. With a = 8, b = 10 and A = 30 degrees the height is 5. If a is shorter than 5 no triangle closes. If a is exactly 5 there is one, with a right angle at B. If a is between 5 and 10 there are two. If a is 10 or more there is one. When the given angle is 90 degrees or more, a has to be longer than b for any triangle at all, and then there is exactly one. The tool works this out for you and the readout says one triangle or two triangles.
Why are the two answers so different?: Because the law of sines gives the sine of angle B, and two angles below 180 degrees share every sine: B and 180 minus B. With these numbers that is 38.6822 and 141.3178 degrees. Both leave room for angle C, so both close, and they make a long triangle with c at 14.9053 and a short one with c at 2.41526. A textbook question usually asks for both; if a drawing or a measurement tells you which corner is obtuse, that picks one of them.
Which boxes count as side-side-angle?: Any two sides with an angle that sits opposite one of them rather than between them: a and b with A or B, b and c with B or C, a and c with A or C. Type the angle between the two sides instead, C for a and b, and it is the SAS case, which always has exactly one answer; the tool reads which case you gave it and says so above the result.

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.