Finding the third side of a triangle with 2 given side lengths is possible only when additional information—such as an angle, the perimeter, or a special triangle property—is also known. With only two sides, the exact third side cannot be determined, but its possible range can be calculated using the triangle inequality theorem.
Introduction
A triangle has three sides and three interior angles, and these six measurements are connected by several geometric rules. When two side lengths are known, many people expect one automatic formula to reveal the remaining side. In reality, the correct method depends on what other information is available.
Here's one way to look at it: two sides measuring 6 cm and 8 cm could form a right triangle with a third side of 10 cm, but they could also form many non-right triangles with different third sides. Knowing whether the angle between the two sides is 60°, 90°, or 120° changes the answer.
The most common methods include:
- The triangle inequality theorem when only two sides are known
- The Pythagorean theorem for right triangles
- The law of cosines when two sides and the included angle are known
- The law of sines for certain side-and-angle combinations
- Additional information such as perimeter, area, or an isosceles relationship
The Key Rule: Two Sides Alone Are Not Enough
Suppose the known sides are $a$ and $b$, and the unknown side is $c$. Unless the triangle has a special property or another measurement is provided, there is no single value for $c$ That's the part that actually makes a difference..
The third side must satisfy the triangle inequality theorem:
$ |a-b| < c < a+b $
This means:
- $c$ must be greater than the absolute difference between the known sides.
- $c$ must be less than the sum of the known sides.
- Equality is not allowed in a genuine triangle. If $c=a+b$, the three points form a straight line rather than a triangle.
Example
If the known sides are 7 cm and 11 cm:
$ 11-7 < c < 11+7 $
$ 4 < c < 18 $
The third side can be any length between 4 cm and 18 cm, but it cannot equal either endpoint. That's why, its exact length remains unknown without more information.
Method 1: Use the Pythagorean Theorem for a Right Triangle
If the triangle is known to be a right triangle, the Pythagorean theorem is usually the fastest method:
$ a^2+b^2=c^2 $
Here, $c$ represents the hypotenuse, the side opposite the 90° angle and the longest side of the triangle.
When the Two Known Sides Are the Legs
If both shorter sides forming the right angle are known, calculate:
$ c=\sqrt{a^2+b^2} $
For legs of 5 cm and 12 cm:
$ c=\sqrt{5^2+12^2} $
$ c=\sqrt{25+144}=\sqrt{169}=13 $
The third side is 13 cm.
When One Known Side Is the Hypotenuse
If the hypotenuse $h$ and one leg $a$ are known, rearrange the formula:
$ b=\sqrt{h^2-a^2} $
For a hypotenuse of 17 cm and one leg of 8 cm:
$ b=\sqrt{17^2-8^2} $
$ b=\sqrt{289-64}=\sqrt{225}=15 $
The missing leg is 15 cm That's the part that actually makes a difference..
Always verify that the side identified as the hypotenuse is the longest given side. Using a shorter side as the
