Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: The Law of Sines, a/sin A = b/sin B = c/sin C, solves triangles given ASA, AAS or SSA, and the Law of Cosines, c² = a² + b² − 2ab cos C, solves SAS and SSS. Triangle area is ½ab sin C or Heron’s formula, and the SSA case can produce zero, one or two triangles.
Standards: G-SRT.D.9, G-SRT.D.10, G-SRT.D.11 · Level: Grade 12 Precalculus
How do you find the area of a triangle from two sides and an angle?
If sides a and b enclose angle C, the height to side a is b sin C, so the area is ½ab sin C. When all three sides are known, use Heron’s formula.
- Area = ½ab sin C (two sides and the included angle).
- Heron: with s = (a + b + c)/2, Area = √(s(s − a)(s − b)(s − c)).
Example: sides 9, 10 and 17 give s = 18 and area = √(18 · 9 · 8 · 1) = √1296 = 36.
When do you use the Law of Sines and the Law of Cosines?
| Given | Use | Formula |
|---|---|---|
| ASA, AAS, or SSA | Law of Sines | a/sin A = b/sin B = c/sin C |
| SAS or SSS | Law of Cosines | c2 = a2 + b2 − 2ab cos C |
With a = 8, b = 11 and C = 52°: c2 = 64 + 121 − 176 cos 52° ≈ 76.64, so c ≈ 8.75.
What is the ambiguous case of the Law of Sines?
Given angle A and sides a and b (SSA), compare a with the height h = b sin A. If a < h, no triangle exists. If a = h, one right triangle. If h < a < b, two triangles. If a ≥ b, one triangle.
In the figure, h = 10 sin 30° = 5 and 5 < 6 < 10, so two triangles fit. The second angle is B2 = 180° − B1.
How do bearings work in navigation problems?
A bearing is the angle measured clockwise from north. Draw a north line at every turning point, find the interior angle of the triangle from the bearings, then apply the Law of Cosines or Sines.
The interior angle at the turn is 180° − (115° − 40°) = 105°. Then d2 = 202 + 152 − 2(20)(15) cos 105° ≈ 780.29, so d ≈ 27.9 miles.
Using the Law of Sines for SAS. You need a side and its opposite angle for a pair, and SAS does not give one. Use the Law of Cosines first.
Make a sketch and label the sides a, b, c opposite angles A, B, C. Check that the longest side faces the largest angle.
- Area = ½ab sin C; Heron: √(s(s − a)(s − b)(s − c)).
- Law of Sines for ASA, AAS, SSA; Law of Cosines for SAS, SSS.
- SSA can give no triangle, one right triangle, one triangle, or two triangles.
- A bearing is measured clockwise from north.
- The largest angle is opposite the longest side.
Law of Sines, Law of Cosines and Area: frequently asked questions
When should you use the Law of Sines instead of the Law of Cosines?
Use the Law of Sines when you know a side and its opposite angle (ASA, AAS, SSA). Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS).
What is the ambiguous case in the Law of Sines?
With two sides and a non-included angle (SSA), the data can describe zero, one or two triangles. Compare the side opposite the angle with b sin A, and if two triangles fit, use B and 180° − B.
How does Heron’s formula find the area?
It uses only the three side lengths. Compute the semiperimeter s = (a + b + c)/2, then take the square root of s(s − a)(s − b)(s − c). No angle or height is needed.
Why does the area equal ½ab sin C?
Taking side a as the base, the height is b sin C because the height is the vertical leg of a right triangle with hypotenuse b and angle C. Then area = ½ × base × height.
What is a bearing?
A bearing is a direction given as an angle measured clockwise from north, usually written with three digits. A bearing of 090° is east, 180° is south and 270° is west.
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