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Solving SAS Triangles — Definition, Formula & Examples

Solving SAS triangles means finding all unknown sides and angles of a triangle when you know two sides and the angle between them (Side-Angle-Side). You use the Law of Cosines first to find the third side, then the Law of Sines or the Law of Cosines again to find the remaining angles.

Given a triangle with two known side lengths aa and bb and the included angle CC (the angle formed between those two sides), the triangle is fully determined. The unknown side cc is computed via the Law of Cosines, and the remaining angles AA and BB are found using the Law of Sines or by applying the Law of Cosines a second time, with the constraint that A+B+C=180°A + B + C = 180°.

Key Formula

c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab\cos(C)
Where:
  • aa = One of the two known side lengths
  • bb = The other known side length
  • CC = The included angle (between sides a and b)
  • cc = The unknown side opposite angle C

How It Works

Start by identifying the two known sides and the angle between them — that included angle is the key to SAS. Apply the Law of Cosines to calculate the third side opposite the known angle. Next, use the Law of Sines to find one of the two remaining angles. Finally, subtract the two known angles from 180° to get the last angle. Because the given angle is included between the two known sides, SAS always produces exactly one triangle — there is no ambiguous case.

Worked Example

Problem: In triangle ABC, side a = 8, side b = 5, and the included angle C = 60°. Find side c and angles A and B.
Step 1: Apply the Law of Cosines to find side c: Substitute the known values into the Law of Cosines formula.
c2=82+52−2(8)(5)cos⁡(60°)=64+25−80(0.5)=49c^2 = 8^2 + 5^2 - 2(8)(5)\cos(60°) = 64 + 25 - 80(0.5) = 49
Step 2: Take the square root: Since c² = 49, take the positive square root.
c=49=7c = \sqrt{49} = 7
Step 3: Use the Law of Sines to find angle A: Set up the proportion using the Law of Sines with side a and side c.
sin⁡Aa=sin⁡Cc  ⟹  sin⁡A=8sin⁡60°7=8(0.8660)7≈0.9897\frac{\sin A}{a} = \frac{\sin C}{c} \implies \sin A = \frac{8 \sin 60°}{7} = \frac{8(0.8660)}{7} \approx 0.9897
Step 4: Solve for angle A: Take the inverse sine. Since A must be less than 180° and is opposite the longest side, check whether it could be obtuse. Here sin A ≈ 0.9897 gives A ≈ 81.8° (acute) or 98.2° (obtuse). Since a = 8 is the longest side, A should be the largest angle, and 98.2° + 60° = 158.2° leaves room for angle B, so A ≈ 98.2°.
A≈98.2°A \approx 98.2°
Step 5: Find angle B: Subtract the known angles from 180°.
B=180°−60°−98.2°=21.8°B = 180° - 60° - 98.2° = 21.8°
Answer: Side c = 7, angle A ≈ 98.2°, and angle B ≈ 21.8°.

Another Example

Problem: In triangle PQR, side p = 10, side q = 6, and the included angle R = 40°. Find side r.
Step 1: Apply the Law of Cosines: Use the formula with the two known sides and the included angle.
r2=102+62−2(10)(6)cos⁡(40°)=100+36−120(0.7660)≈44.08r^2 = 10^2 + 6^2 - 2(10)(6)\cos(40°) = 100 + 36 - 120(0.7660) \approx 44.08
Step 2: Take the square root: Find r by taking the positive square root of the result.
r≈44.08≈6.64r \approx \sqrt{44.08} \approx 6.64
Answer: Side r ≈ 6.64.

Why It Matters

SAS triangles appear constantly in trigonometry and precalculus courses, especially in problems involving navigation, surveying, and force vectors. Engineers use SAS solutions when they know two distances from a point and the angle between the directions. Mastering this method also prepares you for more advanced topics like the area formula 12absin⁡C\frac{1}{2}ab\sin C, which relies on the same SAS information.

Common Mistakes

Mistake: Using the Law of Cosines with the wrong angle — plugging in an angle that is not between the two known sides.
Correction: The angle in the formula must be the included angle, the one formed between the two given sides. Double-check your triangle labeling before substituting.
Mistake: Forgetting to consider the obtuse angle possibility when using the Law of Sines to find the remaining angles.
Correction: The inverse sine function only returns acute angles. If the angle you are solving for could be obtuse (opposite the longest side), you must check whether 180° minus the calculator's answer is the correct solution.

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