30 60 90 Triangle Calculator

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  30 60 90 Triangle Calculator  

The 30 60 90 Triangle Calculator is a free and easy-to-use online tool that helps you quickly calculate all sides and properties of a 30-60-90 triangle based on a single known value. Whether you’re working with the base, perpendicular (height), hypotenuse, altitude, area, or perimeter, simply enter any one value, and the calculator will instantly determine all the remaining dimensions with accuracy and ease.

It also provides clear, step-by-step explanations, making it ideal for learning geometric concepts, verifying your work, or using it as a quick reference tool—conveniently available on CalculationClub.com.

How to Use the Online 30-60-90 Triangle Calculator

1. Enter Any One Value
Start by entering any one known value of the triangle—such as side a (short leg), side b (long leg), side c (hypotenuse), h (altitude), Area, or Perimeter. The calculator only needs one input to compute the rest.

2. Choose Decimal Precision and Units
Select the number of decimal places you’d like the results to display (e.g., 3), and choose your preferred unit of measurement, such as meters (m) :

Then, choose your preferred unit of measurement from the following options:

1. Meters (m)
2. Centimeters (cm)
3. Millimeters (mm)
4. Yards (yd)
5. Feet (ft)
6. Inches (in)

3. View Instant Results
As soon as you enter a value, the calculator automatically computes all the missing dimensions of the triangle using the fixed 30-60-90 side ratios.

4. Show or Hide Steps
Click “Show Steps” to view a detailed explanation of how each value was calculated. You can also click “Hide Steps” for a cleaner, summary-only view.

5. Reset for a New Calculation
Click the “Reset” button to clear all fields and enter a new set of values for another triangle calculation.

30 60 90 Triangle Calculator

What is a 30 60 90 Triangle?

A 30-60-90 triangle is one of the most important types of right-angled triangles in geometry. It is called a special triangle because its angle measures are always fixed—30 degrees, 60 degrees, and 90 degrees. Because it includes a 90° angle, it’s always a right triangle. That’s why we can apply the Pythagoras Theorem to this triangle.

Sides of the 30-60-90 Triangle

In every 30-60-90 triangle, the three sides are always in a fixed ratio. Once you know the length of any one side, you can calculate the other two using this ratio.

Let’s assign the sides their standard names:

• Short Leg (opposite 30° angle): a
• Long Leg (opposite 60° angle): a√3
• Hypotenuse (opposite 90° angle): 2a

Explanation of Each Side:

1. Short Leg (Perpendicular): This side is directly opposite the 30° angle. It is the smallest side in the triangle and serves as the base for determining the others.

2. Long Leg (Base): This side lies opposite the 60° angle. It is always √3 times longer than the short leg.

3. Hypotenuse: This is the side opposite the right angle (90°) and is always the longest side of the triangle. It is twice as long as the short leg.

Short leg : Long leg : Hypotenuse = 1: 3 :2

Altitude of a 30-60-90 Triangle:

In a triangle, an altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension). In a 30-60-90 triangle, we often draw the altitude from the right angle (90°) to the hypotenuse. This splits the triangle into two smaller right-angled triangles.

Formula to Calculate Altitude:

• Short leg = a
• Long leg = a√3
• Hypotenuse = 2a

Step-by-Step Explanation:

Step 1: Area Expressions- We can calculate the area of a triangle in two ways:

a. Using the base and height (altitude):

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

$\text{Area} = \frac{1}{2} \times c \times h$

Where:

• • c = Hypotenuse (for formula it is base)
    
  • h = Altitude from the right angle to the hypotenuse (for formula it is height)

b. Using the two legs (short leg a and long leg b ):

$\text{Area} = \frac{1}{2} \times a \times b$

Where:

• • a = Short leg (Perpendicular)
    
  • $b$ = Long leg (Base)

Step 2: Solving for Altitude (h): By setting both area expressions equal to each other, we can solve for the altitude (h).

$\text{Area}$ = $\text{Area}$

$\frac{1}{2} \times c \times h$= $\frac{1}{2} \times a \times b$

Simplifying this equation:

h = $\frac{a \times b}{c} $

​Where:

• c = Hypotenuse (for formula it is base)
  
• h = Altitude from the right angle to the hypotenuse (for formula it is height)
• a = Short leg (Perpendicular)
  
• $b$ = Long leg (Base)

Step 3: Substituting Values

Now, let’s substitute the values for the sides of the triangle:

• a = x (short leg)
• b = x√3 (long leg)
• c = 2x (hypotenuse)

$h = \frac{x \times x\sqrt{3}}{2x}$​​​

Simplifying the equation:

$h = \frac{x \times \sqrt{3}}{2}$​​

So, the altitude (h) from the right angle to the hypotenuse is:

$\mathbf{h = \frac{x\sqrt{3}}{2}}$​​

What is the Pythagorean Theorem?

The Pythagorean Theorem states:

$\text{(Hypotenuse)}^2 = (\text{Base})^2 + (\text{Height})^2 $

or c² = a² + b²

This theorem applies only to right-angled triangles, and since the 30-60-90 triangle has a 90° angle, it qualifies.

Summary of a 30-60-90 Triangle:

Key Properties of 30 60 90 Triangle:

1. It’s always a right triangle because it includes a 90° angle; that’s why we can apply the Pythagoras Theorem to this triangle.
2. The sides have fixed ratios based on the angles.

Side Length Ratios of 30 60 90 Triangle:

A 30-60-90 triangle has a fixed ratio of sides: 1 : √3 : 2

If the shortest side (opposite the 30° angle) is of length a, then:

1. The side opposite the 60° angle is a√3.
2. The hypotenuse (opposite the 90° angle) is 2a.
3. If any one side is known, all other sides can be calculated.
4. Pythagorean Theorem holds true because the triangle has a right angle

Example of 30 60 90 Triangle:

If the side opposite 30° is 5cm:

• Side opposite 60° = 5√3 ≈ 8.66cm
• Hypotenuse = 10cm

Formulas: 30°-60°-90° Triangle

Side:

In a 30°-60°-90° triangle, the sides are always in the ratio:

$\text{Short Leg} : \text{Long Leg} : \text{Hypotenuse} = 1 : \sqrt{3} : 2$

Let the short leg be $x$, then:

Short Leg = $x$

Long Leg = $x\sqrt{3}$

Hypotenuse = $2x$

🏔️ Altitude (h) from Right Angle to Hypotenuse

Using the area expression:
$\text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times c \times h$

Solving for $h$:
$h = \frac{a \times b}{c} = \frac{x \cdot x\sqrt{3}}{2x} = \frac{x\sqrt{3}}{2}$

So,
$\mathbf{h = \frac{x\sqrt{3}}{2}}$

📏 Perimeter (P)

Add all sides:
$P = x + x\sqrt{3} + 2x = x(3 + \sqrt{3})$
$\mathbf{P = x(3 + \sqrt{3})}$

🧮 Area (A)

Using the short leg and long leg as base and height:
$A = \frac{1}{2} \cdot x \cdot x\sqrt{3} = \frac{x^2\sqrt{3}}{2}$
$\mathbf{A = \frac{x^2\sqrt{3}}{2}}$

Problem: 30°-60°-90° Triangle

Example.1 = Given a 30°-60°-90° triangle with the short leg $x = 5$ units, we need to find the long leg, hypotenuse, altitude, perimeter, and area of the triangle.

Solution: The sides of a 30°-60°-90° triangle follow the ratio $1 : \sqrt{3} : 2$, with the short leg representing $x$. Since the short leg is given as $x = 5$, we can calculate the other properties.

The long leg is $x\sqrt{3}$, which gives:

$\text{Long Leg} = 5\sqrt{3}$ $ \approx 8.66 \text{ units}$

The hypotenuse is always twice the length of the short leg, so:

$\text{Hypotenuse} = 2x = 2 \times 5 = 10 \text{ units}$

The altitude $h$ from the right angle to the hypotenuse can be calculated using the formula:

$h = \frac{x\sqrt{3}}{2}$ $ = \frac{5\sqrt{3}}{2}$ $ \approx 4.33 \text{ units}$

The perimeter $P$ is the sum of all three sides:

$P = x + x\sqrt{3} + 2x$ $ = 5 + 8.66 + 10$ $ = 23.66 \text{ units}$

Finally, the area $A$ of the triangle can be calculated using the base (short leg) and height (long leg):

$A = \frac{1}{2} \times 5 \times 5\sqrt{3}$ $ = \frac{1}{2} \times 5 \times 8.66$ $ \approx 21.65 \text{ square units}$

Example.2 = Given a 30°-60°-90° triangle with the long leg $9$ units, we need to find the short leg, hypotenuse, altitude, perimeter, and area of the triangle.

Solution: The sides of a 30°-60°-90° triangle follow the ratio $1 : \sqrt{3} : 2$, with the long leg representing $x\sqrt{3}$. To find the short leg $x$, solve:

$x = \frac{9}{\sqrt{3}}$ $ \approx 5.2 \text{ units}$

Now we can calculate the other properties.

The short leg is:

$\text{Short Leg} = x = \frac{9}{\sqrt{3}}$ $ \approx 5.2 \text{ units}$

The hypotenuse is twice the short leg:

$\text{Hypotenuse} = 2x = 2 \times \frac{9}{\sqrt{3}}$ $ \approx 10.4 \text{ units}$

The altitude $h$ from the right angle to the hypotenuse:

$h = \frac{x\sqrt{3}}{2} = \frac{9}{2} = 4.5 \text{ units}$

The perimeter $P$ is the sum of all three sides:

$P = x + x\sqrt{3} + 2x = 5.2 + 9 + 10.4 = 24.6 \text{ units}$

Finally, the area $A$ of the triangle using base (short leg) and height (long leg):

$A = \frac{1}{2} \times x \times x\sqrt{3} = \frac{1}{2} \times 5.2 \times 9$ $ \approx 23.4 \text{ square units}$

Example.3 = Given a 30°-60°-90° triangle with the hypotenuse 12 units, we need to find the short leg, long leg, altitude, perimeter, and area of the triangle.

Solution: In a 30°-60°-90° triangle, the hypotenuse is $2x$. Solving for $x$:

$x = \frac{12}{2} = 6$

Now we can calculate the other properties.

The short leg is:

$\text{Short Leg}$ $ = x = 6 \text{ units}$

The long leg is:

$\text{Long Leg}$ $ = x\sqrt{3} = 6\sqrt{3}$ $ \approx 10.39 \text{ units}$

The altitude $h$ from the right angle to the hypotenuse:

$h = \frac{x\sqrt{3}}{2} $ $= \frac{6\sqrt{3}}{2}$ $ = 3\sqrt{3} \approx 5.20 \text{ units}$

The perimeter $P$ is the sum of all three sides:

$P = x + x\sqrt{3} + 2x$ $ = 6 + 10.39 + 12$ $ = 28.39 \text{ units}$

The area $A$ of the triangle using base and height:

$A = \frac{1}{2} \times 6 \times 10.39$ $ \approx 31.17 \text{ square units}$

Example.4 = Given a 30°-60°-90° triangle where the altitude from the right angle to the hypotenuse is $h = 6$ units, find the short leg, long leg, hypotenuse, perimeter, and area.

Solution: The altitude from the right angle to the hypotenuse in a 30°-60°-90° triangle is given by:

$h = \frac{x\sqrt{3}}{2}$

Solving for $x$:

$x = \frac{2h}{\sqrt{3}}$ $ = \frac{2 \times 6}{\sqrt{3}} \approx 6.93$

Now calculate the other properties.

Short leg:

$\text{Short Leg}$  $= x \approx 6.93 \text{ units}$

Long leg:

$\text{Long Leg} = x\sqrt{3} \approx 6.93 \times \sqrt{3}$ $ = 12 \text{ units}$

Hypotenuse:

$\text{Hypotenuse}$ $ = 2x \approx 2 \times 6.93$ $ = 13.86 \text{ units}$

Perimeter:

$P = x + x\sqrt{3} + 2x \approx 6.93 + 12 + 13.86$ $ = 32.79 \text{ units}$

Area:

$A = \frac{1}{2} \times 6.93 \times 12$ $ \approx 41.58 \text{ square units}$

Example.5 = Given a 30°-60°-90° triangle with a perimeter of $P = 24$ units, find the short leg, long leg, hypotenuse, altitude, and area.

Solution: The perimeter in terms of $x$ is:

$P = x + x\sqrt{3} + 2x = 3x + x\sqrt{3}$

Factor out $x$:

$P = x(3 + \sqrt{3})$

Solving for $x$:

$x = \frac{24}{3 + \sqrt{3}}$ $ \approx 5.12$

Now we compute the remaining properties.

Short leg:

$\text{Short Leg} = x \approx 5.12 \text{ units}$

Long leg:

$\text{Long Leg} = x\sqrt{3} \approx 5.12 \times \sqrt{3} = 8.87 \text{ units}$

Hypotenuse:

$\text{Hypotenuse}$ $ = 2x = 10.24 \text{ units}$

Altitude:

$h = \frac{x\sqrt{3}}{2} \approx \frac{8.87}{2}$ $ = 4.43 \text{ units}$

Area:

$A = \frac{1}{2} \times 5.12 \times 8.87 \approx 22.72 \text{ square units}$

Example.6 = Given a 30°-60°-90° triangle with an area of $A = 27$ square units, find the short leg, long leg, hypotenuse, altitude, and perimeter.

Solution: The area in terms of $x$ is:

$A = \frac{1}{2} \times x \times x\sqrt{3}$ $= \frac{x^2\sqrt{3}}{2}$

Solving for $x$:

$x^2 = \frac{2A}{\sqrt{3}} = \frac{2 \times 27}{\sqrt{3}}$ $ \approx 31.18$

$x \approx \sqrt{31.18} \approx 5.58$

Now compute the other properties.

Short leg:

$\text{Short Leg} = x \approx 5.58 \text{ units}$

Long leg:

$\text{Long Leg}$ $ = x\sqrt{3} \approx 9.66 \text{ units}$

Hypotenuse:

$\text{Hypotenuse}$ $ = 2x \approx 11.16 \text{ units}$

Altitude:

$h = \frac{x\sqrt{3}}{2} \approx \frac{9.66}{2}$ $ = 4.83 \text{ units}$

Perimeter:

$P = x + x\sqrt{3} + 2x = 5.58 + 9.66 + 11.16 = 26.4 \text{ units}$

Frequently Asked Questions (FAQs) on 30-60-90 Triangle Calculator

1. What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90°. The sides of this triangle have a consistent ratio of 1 : √3 : 2.

2. How does the 30-60-90 triangle calculator work?
The calculator uses the known side ratios of the triangle. If you input one side (short leg, long leg, or hypotenuse), the calculator will automatically compute the other two using the 1 : √3 : 2 ratio.

3. Which side is which in a 30-60-90 triangle

1. The short leg is opposite the 30° angle.
2. The long leg is opposite the 60° angle.
3. The hypotenuse is opposite the right angle (90°) and is the longest side.

4. What is the ratio of the sides in a 30-60-90 triangle?
The side lengths follow the ratio:
Short leg : Long leg : Hypotenuse = 1 : √3 : 2

5. Can I use any unit for input?
Yes, as long as you use the same unit for all sides. The calculator preserves the unit in its output.

6. Is this triangle always a right triangle?
Yes, by definition, a 30-60-90 triangle is always a right triangle because it includes a 90° angle.

7. What if I enter the long leg or hypotenuse instead of the short leg?
The calculator can work with any one side. If you provide the long leg or hypotenuse, it will reverse-calculate the short leg first and then use the ratio to find the other sides.

8. Can this calculator show steps or formulas used?
Some versions of the calculator display the formulas used in the background. If yours doesn’t, you can manually apply:

1. Long leg = Short leg × √3
2. Hypotenuse = Short leg × 2

9. Why is this triangle useful in geometry or trigonometry?
It simplifies calculations due to its predictable ratios and is commonly used in trigonometric problems, architectural design, and standardized tests.

10. Can I use this calculator for solving real-life problems?
Absolutely. It’s helpful for solving construction problems, ramp angles, or any application involving a 30-60-90 triangle.

Final Thoughts: The 30 60 90 Triangle Calculator is a simple yet powerful tool that instantly solves for all sides and properties of the triangle from just one known value. It’s perfect for students, teachers, and professionals alike—fast, accurate, and available anytime on CalculationClub.com.

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