What is the area formula for two sides and the included angle?

Area = ½ab sin C: half the product of the two sides times the sine of the angle between them. With 5, 7 and 60 degrees that is ½ × 5 × 7 × 0.866025, which is 15.1554. It comes from the base and height formula: b sin C is the height from corner A down to side a's line, so ½ × a × (b sin C) is half a base times its height. Use whichever pair of sides you know with the angle between that pair, b and c with A, or a and c with B.

Which angle is the included angle?

The one where the two given sides meet, which is always the angle opposite the third, missing side. Sides a and b meet at C, b and c at A, a and c at B. If the angle you have is opposite one of your two sides instead, it is the side-side-angle case, and the tool switches to it: that case can have no triangle, one or two, so it gets its own page.

Can I type the angle in radians?

Yes. Switch Angles in to Radians and the box reads radians: 60 degrees is about 1.0472. Anything already typed is converted as you switch, so the triangle stays the same, and the angles come back in radians.

SAS Triangle Calculator

Give two sides and the angle between them to get the third side by the law of cosines, the other two angles, the area as half ab sin C and each height. The page opens on a = 5, b = 7 and C = 60 degrees. The law of cosines gives c squared as 25 plus 49 minus 2 times 5 times 7 times the cosine of 60, which is 39, so c is 6.245; the same law on the finished triangle gives A as 43.8979 degrees and B as 76.1021 degrees. The area is half of 5 times 7 times the sine of 60, which is 15.1554, and the perimeter is 18.245. Change any of the three boxes and the rest follows, and the figure is redrawn to scale. Side-angle-side always has exactly one answer, so there is no ambiguous case here.

Solvedone triangle

SASa 5 b 7 c 6.245Degrees

a 5   b 7   c 6.245
A 43.8979°   B 76.1021°   C 60°
area 15.1554     perimeter 18.245
height to a 6.06218   to b 4.33013   to c 4.85363
acute, scalene

law of cosines for the third side, then the law of cosines again for the angles

A (°)

B (°)

C (°)

Angles in

43.8979°

76.1021°

60°

Area

15.1554

Two sides and the angle between them, so a, b and C, or b, c and A, or a, c and B. The third side comes from the law of cosines, and the two remaining angles from the same law on the finished triangle.

SAS: 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 area formula for two sides and the included angle?: Area = ½ab sin C: half the product of the two sides times the sine of the angle between them. With 5, 7 and 60 degrees that is ½ × 5 × 7 × 0.866025, which is 15.1554. It comes from the base and height formula: b sin C is the height from corner A down to side a's line, so ½ × a × (b sin C) is half a base times its height. Use whichever pair of sides you know with the angle between that pair, b and c with A, or a and c with B.
Which angle is the included angle?: The one where the two given sides meet, which is always the angle opposite the third, missing side. Sides a and b meet at C, b and c at A, a and c at B. If the angle you have is opposite one of your two sides instead, it is the side-side-angle case, and the tool switches to it: that case can have no triangle, one or two, so it gets its own page.
Can I type the angle in radians?: Yes. Switch Angles in to Radians and the box reads radians: 60 degrees is about 1.0472. Anything already typed is converted as you switch, so the triangle stays the same, and the angles come back in radians.

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.