The length of the third side of a triangle can be found when you have enough information about the triangle, such as two side lengths and an angle, two angles and one side, or the presence of a right angle. Knowing how to find the length of the third side is useful in geometry, construction, navigation, engineering, and many real-world measurement problems Which is the point..
Introduction: Why the Third Side Matters
In a triangle, the three sides are connected to the three angles. If you know enough information about one triangle, you can calculate the missing side length using mathematical rules. Still, the method depends on what information is given Which is the point..
For example:
- If you know all three sides, you can find the angles.
- If you know two sides and the included angle, you can find the third side.
- If you know one side and two angles, you can find the other sides.
- If you only know two side lengths, you usually cannot find the exact third side unless you also know an angle.
The most common methods for finding the third side include the Pythagorean theorem, the Law of Cosines, and the Law of Sines No workaround needed..
When Can You Find the Third Side?
Before using a formula, it actually matters more than it seems. A triangle has three sides and three angles, and not every set of information is enough to determine the missing side.
You can usually find the third side if you know:
- Two sides and the included angle
- Two angles and one side
- Three sides
- A right triangle with two known sides
You cannot find the exact third side if you only know two side lengths and no angle. In practice, for example, if two sides are 5 and 8, the third side could be many possible values. It must be greater than 3 and less than 13, but it could be 4, 6, 9, 11, or another value in that range The details matter here..
This rule is called the Triangle Inequality Theorem. It states:
The third side of a triangle must be greater than the difference of the other two sides and less than their sum.
So, if two sides are 5 and 8:
8 − 5 < third side < 8 + 5
That means:
3 < third side < 13
Method 1: Using the Pythagorean Theorem
The Pythagorean theorem is used when the triangle is a right triangle, meaning one angle measures exactly 90 degrees. In a right triangle, the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side.
The formula is:
a² + b² = c²
Where:
- a and b are the two shorter sides, called legs
- c is the hypotenuse
Example 1: Finding the Hypotenuse
Suppose a right triangle has legs of length 6 and 8. Find the length of the third side Still holds up..
Use:
a² + b² = c²
Substitute the values:
6² + 8² = c²
36 + 64 = c²
100 = c²
Now take the square root:
c = 10
So, the third side is 10 units.
Example 2: Finding a Missing Leg
Suppose a right triangle has a hypotenuse of 13 and one leg of 5. Find the missing leg.
Use:
a² + b² = c²
Substitute:
5² + b² = 13²
25 + b² = 169
Subtract 25 from both sides:
b² = 144
b = 12
So, the missing side is 12 units.
The Pythagorean theorem is one of the simplest ways to find the third side, but it
only works for right triangles. Practically speaking, if the triangle is not a right triangle, you need a more general method. That method is the Law of Cosines And that's really what it comes down to..
Method 2: Using the Law of Cosines
The Law of Cosines is a powerful tool that works for any triangle, not just right triangles. It is especially useful when you know two sides and the included angle and want to find the third side Practical, not theoretical..
The formula is:
c² = a² + b² − 2ab·cos(C)
Where:
- a and b are the two known sides
- C is the included angle (the angle between sides a and b)
- c is the unknown side opposite angle C
Notice that when angle C is 90 degrees, cos(90°) = 0, and the formula simplifies to the Pythagorean theorem. This shows that the Law of Cosines is a generalization of the Pythagorean theorem.
Example 3: Finding the Third Side with Two Sides and an Included Angle
Suppose a triangle has sides of length 7 and 10, and the angle between them is 60 degrees. Find the third side.
Use:
c² = a² + b² − 2ab·cos(C)
Substitute the values:
c² = 7² + 10² − 2(7)(10)·cos(60°)
c² = 49 + 100 − 140·(0.5)
c² = 149 − 70
c² = 79
Now take the square root:
c ≈ 8.89
So, the third side is approximately 8.89 units.
Example 4: Another Law of Cosines Problem
Suppose a triangle has sides of length 9 and 12, and the included angle is 120 degrees. Find the third side.
Use:
c² = 9² + 12² − 2(9)(12)·cos(120°)
c² = 81 + 144 − 216·(−0.5)
c² = 225 + 108
c² = 333
c ≈ 18.25
So, the third side is approximately 18.25 units Worth knowing..
Notice how the obtuse angle (greater than 90 degrees) increases the length of the third side compared to what you might expect from a right triangle. This is because the cosine of an obtuse angle is negative, which adds to the result rather than subtracting from it.
Method 3: Using the Law of Sines
The Law of Sines is used when you know two angles and one side. Also, since the sum of the angles in any triangle is always 180 degrees, knowing two angles lets you find the third angle immediately. Then, the Law of Sines allows you to find the remaining sides Practical, not theoretical..
The formula is:
a / sin(A) = b / sin(B) = c / sin(C)
Where:
- a, b, and c are the sides of the triangle
- A, B, and C are the angles opposite those sides, respectively
Example 5: Finding Missing Sides with Two Angles and One Side
Suppose a triangle has angles of 40 degrees and 60 degrees, and the side opposite the 40-degree angle is 8 units long. Find the other two sides.
First, find the third angle:
Third angle = 180° − 40° − 60° = 80°
Now apply the Law of Sines:
8 / sin(40°) = b / sin(60°) = c / sin(80°)
Find side b (opposite the 60-degree angle):
b = 8 · sin(60°) / sin(40°)
b ≈ 8 · 0.866 / 0.643
b ≈ 10.78
Find side c (opposite the 80-degree angle):
c = 8 · sin(80°) / sin(40°)
**c ≈ 8
