30 60 90 triangle calculator

What is a 30 60 90 triangle?

A 30 60 90 triangle is a special kind of right triangle that possesses unique properties, making it geometrically significant in mathematics and practical applications. Its angles measure 30°, 60°, and 90°, and this specific ratio of angles ensures certain side proportions. Thanks to these proportions, the 30 60 90 triangle is often used in engineering, architecture, and various calculations.

Features and properties of a 30 60 90 triangle

1. Side proportions:

   • The leg opposite the 30° angle is half the hypotenuse.
   • The leg opposite the 60° angle is $\sqrt{3}$ times half the hypotenuse.

2. Unit ratios:

   • If the hypotenuse length is $c$, the length of the leg opposite the 30° angle will be $\frac{c}{2}$.
   • The length of the leg opposite the 60° angle is $\frac{c \sqrt{3}}{2}$.

Thanks to these clear ratios, any problems involving finding the sides of a 30 60 90 triangle are solved easily and precisely.

Formulas

Now let’s explore how these properties can be used to calculate various triangle parameters.

1. If the leg $a$ (opposite the 30° angle) is known:

• Hypotenuse $c$:

  $$c = 2a$$

• Area $A$:

  $$A = \frac{\sqrt{3}}{2} a^2$$

• Perimeter $P$:

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

2. If the hypotenuse $c$ is known:

• Leg $a$:

  $$a = \frac{c}{2}$$

• Other leg $b$ (opposite the 60° angle):

  $$b = a \cdot \sqrt{3} = \frac{c\sqrt{3}}{2}$$

• Area $A$:

  $$A = \frac{\sqrt{3}}{8} c^2$$

• Perimeter $P$:

  $$P = \left(3 + \sqrt{3}\right) \frac{c}{2}$$

3. If the perimeter $P$ is known:

• Leg $a$:

  $$a = \frac{P}{3 + \sqrt{3}}$$

• Hypotenuse $c$:

  $$c = \frac{2P}{3 + \sqrt{3}}$$

• Area $A$:

  $$A = \frac{\sqrt{3}}{2} \left(\frac{P}{3 + \sqrt{3}}\right)^2$$

4. If the area $A$ is known:

• Leg $a$:

  $$a = \sqrt{\frac{2A}{\sqrt{3}}}$$

• Hypotenuse $c$:

  $$c = 2a = 2\sqrt{\frac{2A}{\sqrt{3}}}$$

• Perimeter $P$:

  $$P = (3 + \sqrt{3}) \sqrt{\frac{2A}{\sqrt{3}}}$$

Examples

Example 1: Known leg $a = 4$

1. Hypotenuse $c$:

   $$c = 2a = 2 \cdot 4 = 8$$

2. Area $A$:

   $$A = \frac{\sqrt{3}}{2} a^2 = \frac{\sqrt{3}}{2} \cdot 4^2 = \frac{\sqrt{3}}{2} \cdot 16 = 8\sqrt{3} \approx 13.86$$

3. Perimeter $P$:

   $$P = (3 + \sqrt{3})a = (3 + \sqrt{3}) \cdot 4 = (3 + 1.732) \cdot 4 \approx 4 \cdot 4.732 \approx 18.93$$

Example 2: Known hypotenuse $c = 10$

1. Leg $a$:

   $$a = \frac{c}{2} = \frac{10}{2} = 5$$

2. Other leg $b$:

   $$b = a \cdot \sqrt{3} = 5 \cdot \sqrt{3} \approx 5 \cdot 1.732 \approx 8.66$$

3. Area $A$:

   $$A = \frac{\sqrt{3}}{8} c^2 = \frac{\sqrt{3}}{8} \cdot 10^2 = \frac{\sqrt{3}}{8} \cdot 100 = 12.5\sqrt{3} \approx 21.65$$

4. Perimeter $P$:

   $$P = \left(3 + \sqrt{3}\right) \frac{c}{2} = \left(3 + 1.732\right) \cdot 5 \approx 4.732 \cdot 5 \approx 23.66$$

Example 3: Known perimeter $P = 30$

1. Leg $a$:

   $$a = \frac{P}{3 + \sqrt{3}} = \frac{30}{3 + 1.732} \approx \frac{30}{4.732} \approx 6.34$$

2. Hypotenuse $c$:

   $$c = \frac{2P}{3 + \sqrt{3}} = \frac{2 \cdot 30}{3 + 1.732} \approx \frac{60}{4.732} \approx 12.68$$

3. Area $A$:

   $$A = \frac{\sqrt{3}}{2} \left(\frac{30}{3 + \sqrt{3}}\right)^2 \approx \frac{\sqrt{3}}{2} \cdot 40.12 \approx 34.81$$

Example 4: Known area $A = 10$

1. Leg $a$:

   $$a = \sqrt{\frac{2A}{\sqrt{3}}} = \sqrt{\frac{2 \cdot 10}{\sqrt{3}}} = \sqrt{\frac{20}{\sqrt{3}}} \approx \sqrt{11.55} \approx 3.39$$

2. Hypotenuse $c$:

   $$c = 2a \approx 2 \cdot 3.39 \approx 6.78$$

3. Perimeter $P$:

   $$P = (3 + \sqrt{3}) a = (3 + 1.732) \cdot 3.39 \approx 4.732 \cdot 3.39 \approx 16.08$$

Frequently asked questions

How to find the leg if the hypotenuse is known?

If the hypotenuse $c$ is known, the leg opposite the 30° angle $a$ is $\frac{c}{2}$, and the leg opposite the 60° angle $b$ is $\frac{c \sqrt{3}}{2}$.

Can this triangle be used in architecture and other fields?

Yes, it is often used in architecture and design due to its stability and simplicity in calculations. The 30 60 90 triangle is also used in various types of layouts, construction, and even in creating three-dimensional figures.

What are the advantages of using this type of triangle?

It allows easy calculations in structure design, ensuring result accuracy.

How to calculate similar values but for a 45 45 90 triangle?

For similar calculations with another type of right triangle - 45 45 90, you can use this calculator.