100 LearnersLast updated on August 5th, 2025
30-60-90 Triangle Formula
In mathematics, the 30-60-90 triangle is a special right triangle that has angles measuring 30 degrees, 60 degrees, and 90 degrees. The side lengths of this triangle are in a specific ratio, which allows us to determine the length of any side if one side length is known. In this topic, we will learn the formulas for the side lengths of 30-60-90 triangles.
List of Math Formulas for 30-60-90 Triangles
The 30-60-90 triangle is a special right triangle with properties that allow us to find the side lengths easily. Let’s learn the formulas that relate the sides of a 30-60-90 triangle.
Math Formula for 30-60-90 Triangle
In a 30-60-90 triangle, the sides are in the ratio 1:√3:2. This means:
- The shortest side, opposite the 30-degree angle, is x.
- The side opposite the 60-degree angle is x√3.
- The hypotenuse, opposite the 90-degree angle, is 2x.
Finding the Shortest Side
To find the shortest side, opposite the 30-degree angle, divide the length of the hypotenuse by 2 or divide the side opposite the 60-degree angle by √3.
Finding the Side Opposite the 60-Degree Angle
To find the side opposite the 60-degree angle, multiply the shortest side by √3 or divide the hypotenuse by 2 and then multiply by √3.
Importance of the 30-60-90 Triangle Formulas
The 30-60-90 triangle formulas are crucial in geometry and trigonometry. Here are some reasons why understanding these formulas is important:
- They help simplify calculations involving right triangles.
- They are essential for solving problems in trigonometry and calculus.
- They provide a quick way to understand relationships between side lengths in special triangles.
Tips and Tricks to Memorize 30-60-90 Triangle Formulas
Students often find these geometric formulas tricky. Here are some tips to master the 30-60-90 triangle formulas:
- Remember the side ratios as 1:√3:2.
- Use simple mnemonics like "Short (1), Longer (√3), Longest (2)".
- Visualize the triangle and practice with different side lengths to reinforce your understanding.
Common Mistakes and How to Avoid Them While Using 30-60-90 Triangle Formulas
Students often make errors when working with 30-60-90 triangles. Here are some common mistakes and tips to avoid them.
Mistake 1
Confusing the Angle Ratios
Students sometimes confuse the angle ratios with the side ratios. Remember, the angles are always 30, 60, and 90 degrees. The side lengths have a ratio of 1:√3:2.
Mistake 2
Using Incorrect Ratios
The side lengths must be in the ratio 1:√3:2. Ensure you apply the correct ratio to find the missing sides.
Mistake 3
Misidentifying the Shortest Side
Students sometimes fail to correctly identify the shortest side, which is opposite the 30-degree angle. Always start with this side as the base for calculations.
Mistake 4
Forgetting to Apply √3
When calculating the side opposite the 60-degree angle, ensure to multiply the shortest side by √3.
Mistake 5
Incorrectly Calculating with the Hypotenuse
The hypotenuse is always twice the length of the shortest side. Double-check your calculations to ensure this relationship holds.
Examples of Problems Using 30-60-90 Triangle Formulas
Problem 1
If the shortest side of a 30-60-90 triangle is 5, what is the length of the hypotenuse?
The hypotenuse is 10.
Explanation
Since the hypotenuse is twice the shortest side, 5 * 2 = 10.
Problem 2
If the hypotenuse of a 30-60-90 triangle is 12, what is the length of the side opposite the 60-degree angle?
The side opposite the 60-degree angle is 6√3.
Explanation
The shortest side is half the hypotenuse, 12 / 2 = 6. The side opposite the 60-degree angle is 6 * √3 = 6√3.
Problem 3
If the side opposite the 60-degree angle is 9√3, what is the shortest side?
The shortest side is 9.
Explanation
Since the side opposite the 60-degree angle is the shortest side multiplied by √3, we solve 9√3/√3 = 9.
Problem 4
A 30-60-90 triangle has a shortest side of 8. What is the length of the side opposite the 60-degree angle?
The side opposite the 60-degree angle is 8√3.
Explanation
The side opposite the 60-degree angle is the shortest side multiplied by √3, so 8 * √3 = 8√3.
Problem 5
If the side opposite the 30-degree angle is 7, what is the hypotenuse?
The hypotenuse is 14.
Explanation
The hypotenuse is twice the length of the shortest side, so 7 * 2 = 14.
FAQs on 30-60-90 Triangle Formulas
1.What is the ratio of the sides in a 30-60-90 triangle?
2.How do you find the hypotenuse in a 30-60-90 triangle?
3.How do you calculate the side opposite the 60-degree angle in a 30-60-90 triangle?
4.What is the shortest side of a 30-60-90 triangle if the hypotenuse is 16?
5.How do you find the shortest side in a 30-60-90 triangle if you know the side opposite the 60-degree angle?
Glossary for 30-60-90 Triangle Formulas
- 30-60-90 Triangle: A special right triangle with angles of 30, 60, and 90 degrees.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
- Shortest Side: The side opposite the 30-degree angle in a 30-60-90 triangle.
- Side Opposite the 60-Degree Angle: The side in a 30-60-90 triangle that is √3 times the shortest side.
- Ratio: The relationship between the lengths of the sides of the triangle, specifically 1:√3:2 for a 30-60-90 triangle.
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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.
