Finding the third side of a triangle given two sides is a fundamental geometry task that can be solved using the triangle inequality, the law of cosines, or the law of sines, and this guide provides a step‑by‑step approach It's one of those things that adds up..
Introduction
When you know the lengths of two sides of a triangle, determining the possible length of the third side involves understanding the relationships that govern triangles. This article explains how to find the third side of a triangle using basic theorems and trigonometric laws, offering clear methods, illustrative examples, and answers to common questions. By the end, you will be able to apply these techniques confidently in academic problems or real‑world scenarios such as construction and navigation Still holds up..
Understanding the Basics
The Triangle Inequality Theorem
The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side. In practical terms, if you have sides a and b, the third side c must satisfy:
- a + b > c
- a + c > b
- b + c > a
These inequalities narrow down the possible range for c but do not give an exact value. To pinpoint c, you need additional information about the angle between the known sides or another geometric relationship.
Law of Cosines
The law of cosines relates the lengths of three sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, and c, and angle γ opposite side c, the formula is:
$c^2 = a^2 + b^2 - 2ab \cos(\gamma)$
When you know a, b, and the included angle γ, you can find the third side directly by plugging the values into the equation and solving for c. This method is especially useful in surveying and physics where angles are measurable Turns out it matters..
Law of Sines
The law of sines states that the ratio of a side length to the sine of its opposite angle is constant for all three sides:
$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$
If you know two sides and a non‑included angle, you can use the law of sines to find the unknown angle first, then apply it again to solve for the third side. This approach is common in navigation and astronomy.
Step‑by‑Step Methods to Find the Third Side
- Identify the known elements – Determine whether you have two sides with the included angle (SAS) or two sides with a non‑included angle (SSA).
- Choose the appropriate law – Use the law of cosines for SAS cases and the law of sines for SSA cases, ensuring the triangle inequality is satisfied.
- Apply the formula – Substitute the known values into the chosen law.
- Solve for the unknown – Rearrange the equation algebraically to isolate the third side.
- Verify the result – Check that the computed side length respects the triangle inequality and that the angles sum to 180°.
Example Using the Law of Cosines
Suppose side a = 7 cm, side b = 10 cm, and the included angle γ = 45°.
- Compute c²: $c^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos(45°)$ $c^2 = 49 + 100 - 140 \times 0.7071$ $c^2 = 149 - 98.99 \approx 50.01$
- Take the square root: $c \approx \sqrt{50.01} \approx 7.07 \text{ cm}$
Thus, the third side is approximately 7.07 cm.
Example Using the Law of Sines
If side a = 5 cm, side b = 8 cm, and angle α opposite side a is 30°:
- Find angle β using the law of sines: $\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}$ $\
