How to Calculate Length in a Triangle?

Calculating the length of a side in a triangle depends primarily on the type of triangle and the information you already possess about its other sides and angles. The most common methods involve the Pythagorean theorem for right-angled triangles and the Law of Sines or Law of Cosines for any type of triangle.

1. For Right-Angled Triangles: The Pythagorean Theorem

A right-angled triangle is a triangle in which one of the angles is exactly 90 degrees. The side opposite the right angle is called the hypotenuse (always the longest side), and the other two sides are called legs.

The Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

• Formula: $a^2 + b^2 = c^2$

How to Use It:

1. Identify the sides: Determine which sides are the legs (a and b) and which is the hypotenuse (c).
2. Substitute known values: Plug in the lengths of the two sides you know into the formula.
3. Solve for the unknown side: Isolate the variable representing the unknown side and calculate its value.

Example:
Imagine a right-angled triangle where the hypotenuse measures 10 units, and one of the shorter sides measures 6 units. To find the length of the other shorter side:

• Let $c = 10$ (hypotenuse)
• Let $a = 6$ (one leg)
• We need to find $b$ (the other leg)

$a^2 + b^2 = c^2$
$6^2 + b^2 = 10^2$
$36 + b^2 = 100$
$b^2 = 100 - 36$
$b^2 = 64$
$b = \sqrt{64}$
$b = 8$

So, the length of the other side is 8 units.

For more detailed information, you can explore resources on the Pythagorean theorem.

2. For Any Triangle: Law of Sines and Law of Cosines

When dealing with triangles that are not right-angled (oblique triangles), you'll need the Law of Sines or the Law of Cosines.

2.1. Law of Sines

The Law of Sines relates the sides of a triangle to the sines of its opposite angles.

• Formula: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
  • Where a, b, c are the side lengths, and A, B, C are the angles opposite those sides, respectively.

When to Use It:

• Angle-Side-Angle (ASA): You know two angles and the included side.
• Angle-Angle-Side (AAS): You know two angles and a non-included side.
• Side-Side-Angle (SSA): You know two sides and a non-included angle (be cautious, this can sometimes lead to ambiguous cases).

Example (AAS):
If you have a triangle with angle $A = 30^\circ$, angle $B = 70^\circ$, and side $a = 5$ units, you can find side $b$:

$\frac{a}{\sin A} = \frac{b}{\sin B}$
$\frac{5}{\sin 30^\circ} = \frac{b}{\sin 70^\circ}$
$b = \frac{5 \times \sin 70^\circ}{\sin 30^\circ}$
$b \approx \frac{5 \times 0.9397}{0.5}$
$b \approx 9.397$ units

Learn more about the Law of Sines.

2.2. Law of Cosines

The Law of Cosines extends the Pythagorean theorem to all triangles. It relates the length of a side to the lengths of the other two sides and the cosine of the angle between them.

• Formulas:
  • $c^2 = a^2 + b^2 - 2ab \cos C$
  • $a^2 = b^2 + c^2 - 2bc \cos A$
  • $b^2 = a^2 + c^2 - 2ac \cos B$

When to Use It:

• Side-Angle-Side (SAS): You know two sides and the included angle.
• Side-Side-Side (SSS): You know all three sides (to find an angle).

Example (SAS):
Consider a triangle with side $a = 7$ units, side $b = 10$ units, and the included angle $C = 60^\circ$. To find side $c$:

$c^2 = a^2 + b^2 - 2ab \cos C$
$c^2 = 7^2 + 10^2 - 2(7)(10) \cos 60^\circ$
$c^2 = 49 + 100 - 140(0.5)$
$c^2 = 149 - 70$
$c^2 = 79$
$c = \sqrt{79}$
$c \approx 8.888$ units

Explore more on the Law of Cosines.

3. Special Triangles

Some triangles have specific side ratios that can simplify calculations:

• 45-45-90 Triangle (Isosceles Right Triangle): The sides are in the ratio $x : x : x\sqrt{2}$. If you know one leg, the other leg is the same length, and the hypotenuse is that length times $\sqrt{2}$.
• 30-60-90 Triangle: The sides are in the ratio $x : x\sqrt{3} : 2x$. If you know the shortest leg (opposite the 30-degree angle), you can find the other leg (opposite the 60-degree angle) by multiplying by $\sqrt{3}$, and the hypotenuse (opposite the 90-degree angle) by multiplying by 2.

Summary of Methods

When calculating lengths in a triangle, always ensure you correctly identify the type of triangle and the specific information you have. Drawing a diagram can be incredibly helpful in visualizing the problem and applying the correct formula.