Math Formula for the Sides of a Triangle
2026-02-28 23:46 Diff
https://brightchamps.com/en-us/math/math-formulas/math-formula-for-the-sides-of-a-triangle

137 Learners

Last updated on September 25, 2025

In geometry, understanding the relationships between the sides of a triangle is crucial. The Pythagorean theorem, the Law of Sines, and the Law of Cosines are key formulas that relate the sides and angles of triangles. In this topic, we will learn the formulas for these fundamental relationships.

List of Math Formulas for the Sides of a Triangle

The relationships between the sides and angles of triangles are expressed through the Pythagorean theorem, the Law of Sines, and the Law of Cosines.

Let’s learn the formulas to calculate the sides of a triangle.

Pythagorean Theorem

The Pythagorean theorem applies to right-angled triangles. It is expressed as: a2 + b2 = c2  where  c  is the hypotenuse, and a  and  b  are the other two sides.

Law of Sines

The Law of Sines relates the sides of a triangle to its angles. It is expressed as: a/sin A = b/sin B =c/sin C where  a, b, c are the sides and  A, B, C  are the opposite angles.

Explore Our Programs

Law of Cosines

The Law of Cosines is useful for calculating an unknown side or angle in any triangle. It is expressed as: c2 = a2 + b2- 2abcos C where  c  is the side opposite angle  C .

Importance of Triangle Side Formulas

Triangle side formulas are fundamental in geometry and real-world applications. Here are some important points about these formulas:

  1. They help in solving geometry problems involving triangles. 
  2. The formulas are essential in fields like architecture, navigation, and physics. 
  3. Understanding these formulas aids in grasping more complex mathematical concepts.

Tips and Tricks to Memorize Triangle Side Formulas

Students often find triangle formulas challenging. Here are some tips to master them: 

  • Use mnemonics or visual aids to remember the Pythagorean theorem and laws of sines and cosines.
  • Practice with different triangle problems to build familiarity. 
  • Create a formula sheet for quick reference and revise it regularly.

Common Mistakes and How to Avoid Them While Using Triangle Side Formulas

Students make errors when applying triangle formulas. Here are some mistakes and how to avoid them:

Problem 1

Find the hypotenuse of a right triangle with sides 3 and 4.

Okay, lets begin

The hypotenuse is 5.

Explanation

Using the Pythagorean theorem: a2 + b2 = c2 

32 + 42 = c2

9 + 16 = c2

25 = c2 

c = 5 

Well explained 👍

Problem 2

In a triangle with sides 7, 9, and angle 45° opposite the side 7, find the third side using the Law of Cosines.

Okay, lets begin

The third side is approximately 5.2.

Explanation

Using the Law of Cosines: c2 = a2 + b2 - 2abcos C 

c2 = 72 + 92 - 2 x 7 x 9 x cos 45°

c2 = 49 + 81 - 126 x 0.7071 

c2 = 130 - 89.131 

c2 = 40.869

c ≈ 6.39

Well explained 👍

Problem 3

In a triangle with angles 30°, 60°, and 90°, and the hypotenuse is 10, find the side opposite the 30° angle.

Okay, lets begin

The side opposite the 30° angle is 5.

Explanation

Using the property of 30-60-90 triangles, the side opposite the 30° angle is half the hypotenuse. Therefore, it is 10/2 = 5.

Well explained 👍

Problem 4

A triangle has sides 5, 12, and angle 90° opposite the side 5. Find the third side.

Okay, lets begin

The third side is 13.

Explanation

Using the Pythagorean theorem: a2 + b2 = c2 

52 + 122 = c2

25 + 144 = c2

169 = c2

c = 13

Well explained 👍

Problem 5

Given a triangle with sides 8, 15, and angle 60° opposite the side 8, find the third side using the Law of Cosines.

Okay, lets begin

The third side is approximately 13.9.

Explanation

Using the Law of Cosines:c2= a2 + b2 - 2abcos C 

c2 = 82 + 152 - 2 x 8 x 15 x cos 60° \

c2 = 64 + 225 - 240 x 0.5 

c2 = 289 - 120 

c2= 169 

c = 13 

Well explained 👍

FAQs on Triangle Side Formulas

1.What is the Pythagorean theorem?

The Pythagorean theorem is: \( a^2 + b^2 = c^2 \) for right-angled triangles.

2.What is the Law of Sines?

The Law of Sines is: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

3.How to use the Law of Cosines?

To use the Law of Cosines, apply: \( c^2 = a^2 + b^2 - 2ab\cos C \).

4.How do you find a missing side in a right triangle?

In a right triangle, use the Pythagorean theorem to find the missing side.

5.What is the relation between angles and sides in triangles?

The angles and sides of triangles are related through trigonometric functions and laws like Sines and Cosines.

Glossary for Triangle Side Formulas

  • Pythagorean Theorem: A formula used to find the sides of right-angled triangles, expressed as \( a^2 + b^2 = c^2 \).
  • Law of Sines: A formula relating the sides and angles of a triangle, expressed as a/sin A = b/sin B = c/sin C.
  • Law of Cosines: A formula to find an unknown side or angle in any triangle, expressed as c2 = a2 + b2 - 2abcos C.
  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • Trigonometric Functions: Functions like sine, cosine, and tangent used to relate angles and sides in triangles.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.