Summarize this article:
100 LearnersLast updated on September 25, 2025
Math Formula for the Sides of a Triangle
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.
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:
- They help in solving geometry problems involving triangles.
- The formulas are essential in fields like architecture, navigation, and physics.
- 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:
Mistake 1
Confusing the Formula for Right Triangles
Students sometimes apply the Pythagorean theorem to non-right triangles. To avoid this, ensure the triangle has a right angle before using the formula.
Mistake 2
Incorrect Angle Measurement
Errors often occur when angles are not measured in the correct units (degrees vs. radians). Always verify the angle measurement before calculation.
Mistake 3
Misapplying the Law of Cosines
Some students use the Law of Cosines incorrectly by not associating the correct sides and angles. Double-check the triangle configuration and ensure proper substitution into the formula.
Mistake 4
Ignoring Triangle Type
Not all triangles are right triangles; students must identify the triangle type to apply the correct formula. Always analyze the given triangle before proceeding.
Mistake 5
Calculation Errors in Trigonometric Functions
Errors can occur when using trigonometric functions. Use a calculator correctly and ensure it's set to the appropriate mode (degree/radian).
Examples of Problems Using Triangle Side Formulas
Problem 1
Find the hypotenuse of a right triangle with sides 3 and 4.
The hypotenuse is 5.
Explanation
Using the Pythagorean theorem: a2 + b2 = c2
32 + 42 = c2
9 + 16 = c2
25 = c2
c = 5
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.
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
Problem 3
In a triangle with angles 30°, 60°, and 90°, and the hypotenuse is 10, find the side opposite the 30° angle.
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.
Problem 4
A triangle has sides 5, 12, and angle 90° opposite the side 5. Find the third side.
The third side is 13.
Explanation
Using the Pythagorean theorem: a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
c = 13
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.
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
FAQs on Triangle Side Formulas
1.What is the Pythagorean theorem?
2.What is the Law of Sines?
3.How to use the Law of Cosines?
4.How do you find a missing side in a right triangle?
5.What is the relation between angles and sides in triangles?
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.
Explore More math-formulas
![Important Math Links Icon]()
Previous to Math Formula for the Sides of a Triangle
![Important Math Links Icon]()
Next to Math Formula for the Sides of a Triangle
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.
