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Solving SSS Triangles — Definition, Formula & Examples

Solving SSS triangles means finding all three angles of a triangle when you know the lengths of all three sides. You use the Law of Cosines to calculate each angle from the given side lengths.

Given a triangle with sides aa, bb, and cc opposite angles AA, BB, and CC respectively, the SSS (Side-Side-Side) case requires determining each interior angle by rearranging the Law of Cosines into the form cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}, then applying the inverse cosine function.

Key Formula

cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}
Where:
  • AA = The angle opposite side $a$
  • aa = The side opposite angle $A$
  • bb = The side opposite angle $B$
  • cc = The side opposite angle $C$

How It Works

Start by using the Law of Cosines to find the angle opposite the longest side — this ensures the angle is computed directly rather than risking ambiguity. Rearrange the formula to isolate the cosine of the unknown angle, then take the inverse cosine. Once you have one angle, you can find the second angle the same way or use the Law of Cosines again. Find the third angle by subtracting the first two from 180°180°. Always check that your three angles sum to 180°180° as a final verification.

Worked Example

Problem: A triangle has sides a = 8, b = 6, and c = 10. Find all three angles.
Step 1: Find angle C (opposite the longest side): Use the Law of Cosines rearranged to solve for angle C, which is opposite side c = 10.
cos⁡C=a2+b2−c22ab=82+62−1022(8)(6)=64+36−10096=096=0\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{8^2 + 6^2 - 10^2}{2(8)(6)} = \frac{64 + 36 - 100}{96} = \frac{0}{96} = 0
Step 2: Compute angle C: Take the inverse cosine to find the angle.
C=cos⁡−1(0)=90°C = \cos^{-1}(0) = 90°
Step 3: Find angle A: Apply the Law of Cosines again for angle A, opposite side a = 8.
cos⁡A=b2+c2−a22bc=36+100−642(6)(10)=72120=0.6  ⟹  A=cos⁡−1(0.6)≈53.13°\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{36 + 100 - 64}{2(6)(10)} = \frac{72}{120} = 0.6 \implies A = \cos^{-1}(0.6) \approx 53.13°
Step 4: Find angle B: Subtract the known angles from 180°.
B=180°−90°−53.13°=36.87°B = 180° - 90° - 53.13° = 36.87°
Answer: The three angles are approximately A ≈ 53.13°, B ≈ 36.87°, and C = 90°.

Another Example

Problem: A triangle has sides a = 5, b = 7, and c = 9. Find all three angles.
Step 1: Find angle C (opposite the longest side, c = 9): Rearrange the Law of Cosines for angle C.
cos⁡C=52+72−922(5)(7)=25+49−8170=−770=−0.1\cos C = \frac{5^2 + 7^2 - 9^2}{2(5)(7)} = \frac{25 + 49 - 81}{70} = \frac{-7}{70} = -0.1
Step 2: Compute angle C: Since cosine is negative, angle C is obtuse.
C=cos⁡−1(−0.1)≈95.74°C = \cos^{-1}(-0.1) \approx 95.74°
Step 3: Find angle A: Apply the Law of Cosines for angle A, opposite side a = 5.
cos⁡A=72+92−522(7)(9)=49+81−25126=105126≈0.8333  ⟹  A≈33.56°\cos A = \frac{7^2 + 9^2 - 5^2}{2(7)(9)} = \frac{49 + 81 - 25}{126} = \frac{105}{126} \approx 0.8333 \implies A \approx 33.56°
Step 4: Find angle B: Subtract from 180°.
B=180°−95.74°−33.56°≈50.70°B = 180° - 95.74° - 33.56° \approx 50.70°
Answer: The three angles are approximately A ≈ 33.56°, B ≈ 50.70°, and C ≈ 95.74°.

Why It Matters

SSS triangle problems appear throughout high school trigonometry and precalculus courses, and they form a core part of standardized test prep. In real-world applications, surveyors and engineers often measure three distances between points and need to determine angles — exactly the SSS scenario. Mastering this method also reinforces your fluency with the Law of Cosines, which carries into physics problems involving vectors and force components.

Common Mistakes

Mistake: Using the Law of Sines to find a second angle after finding the first with the Law of Cosines, then getting the wrong angle because inverse sine cannot distinguish between acute and obtuse angles.
Correction: Either use the Law of Cosines for all angle calculations, or apply the Law of Sines only to find angles you already know must be acute (angles opposite shorter sides).
Mistake: Forgetting to check whether the three sides can actually form a triangle before solving.
Correction: Verify the Triangle Inequality: the sum of any two sides must be greater than the third side. If this fails, no triangle exists.

Related Terms