Mathwords logoMathwords

Solving Triangles — Definition, Formula & Examples

Solving a triangle is the process of finding every unknown side length and angle measure when you are given enough information — typically three pieces, including at least one side. You combine basic angle rules, the Law of Sines, and the Law of Cosines to fill in whatever is missing.

Given a triangle with vertices AA, BB, CC, opposite sides aa, bb, cc, and interior angles α\alpha, β\beta, γ\gamma, solving the triangle means determining all six quantities (a,b,c,α,β,γa, b, c, \alpha, \beta, \gamma) from a sufficient subset of known values. The principal tools are the angle-sum property α+β+γ=180°\alpha + \beta + \gamma = 180°, the Law of Sines asin⁡α=bsin⁡β=csin⁡γ\dfrac{a}{\sin\alpha} = \dfrac{b}{\sin\beta} = \dfrac{c}{\sin\gamma}, and the Law of Cosines c2=a2+b2−2abcos⁡γc^2 = a^2 + b^2 - 2ab\cos\gamma.

Key Formula

asin⁡A=bsin⁡B=csin⁡Candc2=a2+b2−2abcos⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \qquad\text{and}\qquad c^2 = a^2 + b^2 - 2ab\cos C
Where:
  • a,b,ca, b, c = Side lengths opposite angles A, B, C respectively
  • A,B,CA, B, C = Interior angles of the triangle

How It Works

Which method you use depends on what information you start with. If you know two angles and any side (AAS or ASA), find the third angle with α+β+γ=180°\alpha + \beta + \gamma = 180°, then use the Law of Sines to get the remaining sides. If you know two sides and the included angle (SAS), apply the Law of Cosines to find the third side, then the Law of Sines for the remaining angles. If you know all three sides (SSS), use the Law of Cosines to find one angle, then the Law of Sines or the angle-sum rule for the rest. The ambiguous case (SSA) — two sides and an angle opposite one of them — may produce zero, one, or two valid triangles, so extra care is needed.

Worked Example

Problem: In triangle ABC, angle A = 40°, angle B = 75°, and side a = 10. Find angle C, side b, and side c.
Step 1 — Find the third angle: Use the angle-sum property.
C=180°−40°−75°=65°C = 180° - 40° - 75° = 65°
Step 2 — Set up the Law of Sines ratio: Write the common ratio using the known pair a and A.
asin⁡A=10sin⁡40°=100.6428≈15.557\frac{a}{\sin A} = \frac{10}{\sin 40°} = \frac{10}{0.6428} \approx 15.557
Step 3 — Solve for side b: Multiply the ratio by sin B.
b=15.557×sin⁡75°≈15.557×0.9659≈15.03b = 15.557 \times \sin 75° \approx 15.557 \times 0.9659 \approx 15.03
Step 4 — Solve for side c: Multiply the ratio by sin C.
c=15.557×sin⁡65°≈15.557×0.9063≈14.10c = 15.557 \times \sin 65° \approx 15.557 \times 0.9063 \approx 14.10
Answer: Angle C = 65°, side b ≈ 15.03, and side c ≈ 14.10.

Another Example

Problem: In triangle PQR, side p = 8, side q = 11, and the included angle R = 60°. Find side r.
Step 1 — Apply the Law of Cosines: Since you know two sides and the included angle (SAS), use the Law of Cosines.
r2=p2+q2−2pqcos⁡R=82+112−2(8)(11)cos⁡60°r^2 = p^2 + q^2 - 2pq\cos R = 8^2 + 11^2 - 2(8)(11)\cos 60°
Step 2 — Compute: Evaluate each term. Note that cos 60° = 0.5.
r2=64+121−176(0.5)=185−88=97r^2 = 64 + 121 - 176(0.5) = 185 - 88 = 97
Step 3 — Take the square root: Find r.
r=97≈9.85r = \sqrt{97} \approx 9.85
Answer: Side r ≈ 9.85.

Why It Matters

Solving triangles is a core skill in any trigonometry or precalculus course and appears on standardized tests like the SAT and ACT. Surveyors, architects, and navigation professionals use these techniques daily to compute distances and bearings that cannot be measured directly. Mastering it also builds the foundation for vector analysis and physics problems involving forces at angles.

Common Mistakes

Mistake: Using the Law of Sines in an SSA situation without checking for the ambiguous case.
Correction: When you know two sides and a non-included angle, the inverse sine can give two possible angles (θ and 180° − θ). Always test whether both produce a valid triangle before concluding.
Mistake: Forgetting to set your calculator to degree mode (or radian mode) to match the problem's units.
Correction: Before computing any trig function, verify the mode. A sine evaluated in radians when the angle is given in degrees will give a completely wrong answer.

Related Terms