Triangles are one of the most basic geometric shapes, yet they can be somewhat complex for those who are just dipping their toes into the world of geometry. In this article, we’ll discuss methods to calculate the sides of a triangle and become more proficient in solving triangle-related problems.

First, it’s essential to understand some basic terminology and classifications of triangles:

1.Equilateral Triangle: All three sides have equal length.

2.Isosceles Triangle: Two sides have equal length.

3.Scalene Triangle: No sides have equal length.

Now let’s delve into the methods to calculate the sides of a triangle:

1.Pythagorean Theorem (Right Triangles):

The Pythagorean theorem is applicable only to right-angled triangles (triangles that have one 90-degree angle). The theorem states that in a right-angled triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

Formula: a² + b² = c², where ‘a’ and ‘b’ are two shorter sides, and ‘c’ is the hypotenuse or longest side.

Example: If a = 3 and b = 4, then c² = (3²) + (4²) => c² = 9 + 16 => c² = 25 => c = √25 => c = 5

2.Law of Sines (All Types of Triangles):

The law of sines states that the ratio of the length of a side in a triangle to the sine of its opposite angle is constant for all three sides. This means that if we know two angles and one side, or two sides and one non-included angle, we can use this law to determine unknown side lengths.

Formula: 

a / sin(A) = b / sin(B) = c / sin(C)

Example: In a triangle with sides a, b, and c and angles A = 70°, B = 60°, and C = 50°. If side a is known to be 5 units long, the remaining sides can be computed as follows:

b = a * (sin(B) / sin(A)) => b = 5 * (sin(60°) / sin(70°))

c = a * (sin(C) / sin(A)) => c = 5 * (sin(50°) / sin(70°))

3.Law of Cosines (All Types of Triangles):

The law of cosines states the relationship between the lengths of three sides of any triangle and the cosine function of one of its angles. This law is especially useful when given three side lengths or two sides with an included angle.

Formula: 

c² = a² + b² – 2ab * cos(C)

Example: Suppose we have a triangle with sides a = 7 units, b = 9 units, and an included angle C = 120°. We can compute side c as follows:

c² = (7²) + (9²) – 2 * 7 * 9 * cos(120°)

Calculating triangle sides can seem daunting at first, but with knowledge of these key techniques – the Pythagorean theorem, law of sines, and law of cosines – you’ll quickly become proficient in tackling geometry problems involving triangles