Solve with the law of sines
Solved from two angles and a side that isn’t between them (AAS).
| Part | Value | Exact |
|---|---|---|
| Side a given | 10 | |
| Side b | 13.473 | |
| Side c | 15.3209 | |
| Angle A given | 40° | |
| Angle B given | 60° | |
| Angle C | 80° | |
| Area | 66.3414 | |
| Perimeter | 38.7939 | |
| Semiperimeter s | 19.3969 | |
| Height to a (h_a) | 13.2683 | |
| Height to b (h_b) | 9.8481 | |
| Height to c (h_c) | 8.6603 | |
| Inradius r | 3.4202 | |
| Circumradius R | 7.7786 |
Working
- Angle sum. C = 180° − A − B = 180° − 40° − 60° = 80°
- Law of sines. b = a sin B / sin A = 10 × sin 60° / sin 40° = 13.473
- Law of sines. c = a sin C / sin A = 10 × sin 80° / sin 40° = 15.3209
- Perimeter. a + b + c = 10 + 13.473 + 15.3209 = 38.7939
- Heron's formula. s = (a + b + c)/2 = 19.3969; area = √(s(s − a)(s − b)(s − c)) = √(19.3969 × 9.3969 × 5.924 × 4.076) = 66.3414
- Area from two sides. ½ab sin C = ½ × 10 × 13.473 × sin 80° = 66.3414, the same area
- Heights. h = 2 × area / base: h_a = 13.2683, h_b = 9.8481, h_c = 8.6603
- Inradius and circumradius. r = area / s = 3.4202; R = a / (2 sin A) = 7.7786
The law of sines
In any triangle, each side divided by the sine of the angle facing it gives the same number. Written out, a/sin A = b/sin B = c/sin C.
That shared number is the diameter of the circle through the three corners, 2R. So the rule also gives the circumradius, and the calculator lists it as R.
It needs a matched pair. That means a side and the angle facing it, and once you have one pair, any other side or angle you know gives you its partner straight away, because the ratio is shared.
When it applies
| You know | Case | Use |
|---|---|---|
| Two angles and the side between them | ASA | Law of sines, after the angle sum gives the third angle |
| Two angles and another side | AAS | Law of sines, after the angle sum |
| Two sides and an angle not between them | SSA | Law of sines, checking for a second triangle |
| Two sides and the angle between them | SAS | The law of cosines instead |
| Three sides | SSS | The law of cosines instead |
Worked example: two angles and a side
A = 40°, B = 60° and a = 10. Third angle first. C = 180° − 40° − 60° = 80°.
The matched pair is a and A, so 10/sin 40° = 15.5572 is the shared ratio. Then b = 15.5572 × sin 60° = 13.473 and c = 15.5572 × sin 80° = 15.3209.
The area is ½ab sin C = ½ × 10 × 13.473 × sin 80° = 66.3414.
Worked example: two sides and an angle
Wikipedia's article on the rule uses a = 20, c = 24 and C = 40°. Here the pair is c and C, and sin A = 20 sin 40° / 24 = 0.5357, so A = 32.39°.
Could A be 180° − 32.39° = 147.61° instead? No. With C = 40° already in the triangle, the three angles would add up to more than 180° and leave nothing, in fact less than nothing, for B. Side a is shorter than c, so its angle has to be smaller than 40° anyway. Type a = 20, c = 24 and C = 40 above to see the rest.
The ambiguous case
SSA is called the ambiguous case because the answer can be two triangles, one or none. It comes down to how long the side facing the known angle is.
Take A = 35° and b = 8. The height from the top of b down to the base is h = b sin A = 4.5886. Side a has to reach that far.
With a = 3, it can't, and there's no triangle. With a = 6, between h and b, it reaches the base in two places, giving B = 49.8864° or B = 130.1136°. With a = 9, longer than b, it reaches in only one place that makes a triangle, and B = 30.6536°.
When A is 90° or more there's at most one triangle, and only if a is longer than b.
How many triangles in SSA
| Angle A | Side a | Triangles |
|---|---|---|
| Under 90° | Shorter than h | None |
| Under 90° | Equal to h | One, with a right angle at B |
| Under 90° | Between h and b | Two |
| Under 90° | b or longer | One |
| 90° or more | b or shorter | None |
| 90° or more | Longer than b | One |
Why it works
Drop the height from C to side c. It's a leg of two right triangles, so it equals b sin A and also a sin B. Set them equal and divide, and a/sin A = b/sin B. Dropping a height from another corner brings in c the same way.
In an obtuse triangle the height lands outside, on c extended, but sin(180° − B) = sin B, so the rule still holds.
Questions
What is the law of sines formula?
a/sin A = b/sin B = c/sin C, where each side faces the angle with the same letter. Flipped, sin A/a = sin B/b = sin C/c, which is handier for finding an angle.
When should I use the law of sines?
When you know a side and the angle facing it, plus one more side or angle. That covers ASA, AAS and SSA.
Why are there two answers for an angle?
Sine is the same for an angle and its supplement, so sin 30° = sin 150°. The law of sines gives a sine, and both angles may fit.
Can the law of sines find an obtuse angle?
Yes, as the second of the two angles, 180° minus the first. In SAS or SSS, use the law of cosines for the largest angle instead, since cosine tells obtuse from acute.
What does the ratio a/sin A mean?
It's the diameter of the circumcircle, the circle through all three corners. In the 40°, 60°, 10 example it's 15.5572, so R = 7.7786.
Sources
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