When you are given 2 sides of a triangle and need to find the third, the first question is whether you have enough information to determine it. A triangle’s third side is not always possible to calculate from only two side lengths, but it can often be found if you also know an angle, whether the triangle is right, or what type of triangle it is. Understanding the rules of triangles, the triangle inequality theorem, the Pythagorean theorem, and the Law of Cosines will help you solve the problem correctly.
People argue about this. Here's where I land on it.
Introduction: Can You Find the Third Side From Two Sides Alone?
If you are given two sides of a triangle, there may be many different triangles that can be formed using those same two side lengths. The missing side depends on the angle between the known sides or on additional information about the triangle Turns out it matters..
To give you an idea, if a triangle has two known sides of length 5 and 7, the third side cannot be just any number. It must be greater than the difference of the two sides and less than their sum:
7 − 5 < third side < 7 + 5
So:
2 < third side < 12
This means the third side could be 3, 5, 8, 10, or any other length greater than 2 and less than 12, as long as the triangle follows the rules of geometry Which is the point..
To find the exact third side, you usually need one more piece of information, such as:
- The triangle is a right triangle
- The angle between the two known sides is known
- The triangle is equilateral or isosceles
- Another angle is known
- The area of the triangle is known
The Triangle Inequality Theorem
The most important rule to remember when finding the third side of a triangle is the Triangle Inequality Theorem.
This theorem states that the sum of any two sides of a triangle must be greater than the third side.
For a triangle with sides a, b, and c, the following must be true:
- a + b > c
- a + c > b
- b + c > a
If you know two sides, you can use this theorem to find the possible range of the third side Worth keeping that in mind. Still holds up..
Suppose the two known sides are a and b. Then the third side c must satisfy:
|a − b| < c < a + b
This means the third side must be:
- Greater than the absolute difference of the two known sides
- Less than the sum of the two known sides
Example 1: Finding the Possible Range
A triangle has two sides of length 6 and 10. What possible lengths can the third side have?
Use the inequality:
|10 − 6| < c < 10 + 6
4 < c < 16
So the third side must be greater than 4 and less than 16.
This does not give the exact third side, but it tells you which lengths are possible Easy to understand, harder to ignore..
When Two Sides Are Not Enough
It is important to understand that two side lengths alone do not always determine a unique triangle Small thing, real impact..
Imagine two sticks that are 5 units and 8 units long. You can place them together at many different angles. If the angle between them is small, the third side is short. If the angle is large, the third side is long.
That is why knowing only two sides usually gives a range of possible answers, not one exact answer.
To find the exact third side, you need more information. The most common situations are explained below The details matter here..
Finding the Third Side of a Right Triangle
A right triangle has one angle measuring 90 degrees. The side opposite the right angle is called the hypotenuse, and it is always the longest side And that's really what it comes down to..
If you know the two legs of a right triangle, you can find the hypotenuse using the Pythagorean theorem:
a² + b² = c²
Here:
- a and b are the legs
- c is the hypotenuse
Example 2: Finding the Hypotenuse
A right triangle has legs of length 3 and 4. Find the hypotenuse Most people skip this — try not to..
Use:
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = 5
So the third side is 5 units Easy to understand, harder to ignore..
This is the famous 3-4-5 right triangle.
Example 3: Finding a Missing Leg
Suppose a right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg Less friction, more output..
Use:
a² + b² = c²
Let a = 5, c = 13, and solve for b Easy to understand, harder to ignore..
5² + b² = 13²
25 + b² = 169
b² = 144
b = 12
So the missing leg is 12 units.
Finding the Third Side Using the Law of Cosines
If you know two sides of a triangle and the angle between them, you can find the third side using the Law of Cosines.
The Law of Cosines is a general formula that works for any triangle, not just right triangles.
The formula is:
c² = a² + b² − 2ab cos(C)
Here:
- a and b are the two known sides
- C is the included angle between them
- c is the side opposite angle C
The included angle is the
