Have you observed the shape of the slices of pizza we eat? They are almost triangular in shape. The same goes with the set squares that we use for geometry; they are triangular in shape too. Similarly, you will notice several objects around you that are triangular in shape. However, not all triangles are similar. There are some special types of triangles, such as the 30-60-90 triangle. The 30-60-90 triangle is called a special triangle as the angles of this triangle are in a unique ratio of 1:2:3

In this lesson, we will learn all about the 30-60-90 triangle. We will learn about its sides, its area, and the rules that apply to these triangles.

Let's begin!

**Lesson Plan**

**What Is a 30 60 90 Triangle?**

Let's look at the figures below.

What type of triangles do you think these are?

The triangles ABC and PQR are 30-60-90 triangles.

A 30-60-90 triangle is a triangle in which the angles are \(30{^\circ}, 60{^\circ}, \text{ and } 90{^\circ}\).

**What Are the Sides of a 30 60 90 Triangle?**

We will now learn how to find the sides of a 30-60-90 triangle when some of the following details of the triangle are given.

**Base is Given**

Base \(BC\) of the triangle is assumed to be \(a\).

Perpendicular of the triangle \(ABC\) is \(AB\) = \(\dfrac{a}{\sqrt{3}}\)

Hypotenuse of the triangle \(ABC\) is \(AC\) = \(\dfrac{2a}{\sqrt{3}}\)

**Perpendicular is Given**

Perpendicular \(AB\) of the triangle is assumed to be \(a\).

Base of the triangle \(ABC\) is \(BC\) = \(a\sqrt{3}\)

Hypotenuse of the triangle \(ABC\) is \(AC\) = \(2a\)

**Hypotenuse is Given**

Hypotenuse \(AC\) of the triangle is assumed to be \(a\).

Base of the triangle \(ABC\) is \(BC\) = \(\dfrac{a\sqrt{3}}{2}\)

Perpendicular of the triangle \(ABC\) is \(AB\) = \(\dfrac{a}{2}\)

In all the above cases, the ratio of the sides of a 30-60-90 triangle are always in the ratio of \(1 : \sqrt{3} : 2\).

This is also known as the 30-60-90 triangle formula for sides.

\(x : x\sqrt{3} : 2x\) |

**All the sides of a 30-60-90 triangle can be calculated if any one side is given.****Use the trigonometric ratios table of standard angles to write the values of any trigonometric ratio.****If either of the two angles 30 or 60 degrees is given, then the other can be calculated using the angle sum property of the triangle.**

**How to Find the Area of a 30 60 90 Triangle?**

We know that the area of a triangle is \(\dfrac{1}{2} \times \text{base} \times \text{height}\).

The height in a right-angled triangle is the perpendicular of the triangle.

Thus, the area of a right-angle triangle is \(\dfrac{1}{2} \times \text{base} \times \text{perpendicular}\).

Let's learn how to apply this formula to find the area of 30-60-90 triangles.

**Base is Given**

Base \(BC\) of the triangle is assumed to be \(a\).

The perpendicular of the triangle \(ABC\) is \(AB\).

We have learned in the previous section how to find the perpendicular when the base is given.

Let's apply the formula we have learned.

Thus, perpendicular of the triangle = \(\dfrac{a}{\sqrt{3}}\)

Area of the triangle = \(\dfrac{1}{2} \times \text{base} \times \text{perpendicular}\)

Area of the triangle = \(\begin{align}\frac{1}{2} \times a \times \frac{a}{\sqrt{3}}\end{align}\).

Therefore, the area of the 30-60-90 triangle when the base is given as \(a\) is:

\(\dfrac{a^2}{2\sqrt{3}}\) |

- Try to find the ratio of the sides if the angles of the triangles are 37-53-90 degrees.
- What will happen if the perpendicular and the base of the triangle are equal? What are the angles of this triangle?

**30 60 90 Triangle Theorem Proof**

Let's consider an equilateral triangle \(ABC\) with side length equal to \(a\).

Now, draw a perpendicular from vertex \(A\) to side \(BC\) at point \(D\) of the triangle \(ABC\).

The perpendicular in an equilateral triangle bisects the other side.

Triangle \(ABD\) and \(ADC\) are two 30-60-90 triangles.

Both the triangles are similar and right-angled triangles. Hence, we can apply the Pythagoras theorem to find the length AD.

\[\begin{align}\text{AB}^2 &= \text{AD}^2 + \text{BD}^2 \\

a^2 &= \text{AD}^2 + \left(\dfrac{a}{2}\right)^2 \\

a^2 - \left(\dfrac{a}{2}\right)^2 &= \text{AD}^2 \\

\dfrac{3a^2}{4} &= \text{AD}^2 \\

\dfrac{a\sqrt{3}}{2} &= \text{AD} \end{align}\]

\[\begin{align} \text{AD} &= \dfrac{a\sqrt{3}}{2}\\\text{BD} &= \dfrac{a}{2} \\ \text{AB} &= a \end{align}\]

These sides also follows the same ratio \(\dfrac{a}{2} : \dfrac{a\sqrt{3}}{2} : a\)

Multiply by 2 and divide by \(a\).

\(1 : \sqrt{3} : 2\)

This is the proof for the side lengths of a 30-60-90 triangle.

**When To Use 30 60 90 Triangle Rules?**

A 30 60 90 triangle rule can be applied on a right-angled triangle with angles \(30{^\circ}\), \(60{^\circ}\), and \(90{^\circ}\),

When at least one of the sides is given, the others can be calculated using the 30-60-90 triangle rule.

Let's see how we can solve a 30 60 90 triangle using the rule in our next section.

**How To Solve a 30 60 90 Triangle?**

Let's consider an example.

A 30-60-90 triangle has base length equal to 5 units. Let's find the other sides.

First, let's consider that the base angle is \(30{^\circ}\).

If the base angle is \(30{^\circ}\), then the base length is greater than the perpendicular length.

We know that the 30-60-90 triangle sides are \(x\), \(x\sqrt{3}\), and \(2x\), where \(x\) is a constant.

We also know that side length \(BC\) > \(AB\), hence \(\text{BC} = x\sqrt{3}\)

\(\text{5} = x\sqrt{3}\)

\(\dfrac{\text{5}}{\sqrt{3}} = x\)

\(\text{AB} = x\), \(AB\) is the smallest side

\(\text{AB} = \dfrac{\text{5}}{\sqrt{3}}\)

The biggest side of a right-angle triangle is the hypotenuse, \(\text{AC} = 2x\).

\(\text{AC} = 2 \times \dfrac{\text{5}}{\sqrt{3}}\)

\(\text{AC} = \dfrac{\text{10}}{\sqrt{3}}\)

Now, let's consider that the base angle is \(60{^\circ}\).

If the base angle is \(60{^\circ}\), then the base length is smaller than the perpendicular length.

We know that the sides of a 30-60-90 triangle are \(x\), \(x\sqrt{3}\), and \(2x\), where \(x\) is a constant.

We also know that side length \(BC\) < \(AB\), hence, \(\text{BC} = x\)

\(x = \text{5}\)

\(\text{AB} = x\sqrt{3}\)

\(\text{AB} = 5 \times \sqrt{3}\)

\(\text{AB} = 5\sqrt{3}\)

The biggest side of a right angle triangle is the hypotenuse, \(\text{AC} = 2x\).

\(\text{AC} = 2 \times 5\)

\(\text{AC} = 10\)

**Solved Examples**

Example 1 |

Peter and his friend are discussing a problem related to a 30-60-90 triangle.

The hypotenuse of this triangle is given as 20

They need to find the sum of the other two sides of the triangle.

Can you help them solve this problem?

**Solution**

The sides of a 30-60-90 triangle are \(x\), \(x\sqrt{3}\), and \(2x\), where \(x\) is some constant.

The hypotenuse is the biggest side of a right-angle triangle.

Hence, \(2x = 20\)

\(x = 10\)

Peter and his friend want to calculate the sum of the other two sides.

The remaining sides are \(x\) and \(x\sqrt{3}\).

\(x + x\sqrt{3}\)

=\(10 + 10\sqrt{3}\)

=\(10(1 + \sqrt{3})\)

\(\therefore\) Sum is \(10(1 + \sqrt{3})\) |

Example 2 |

Find the length of the hypotenuse of a right-angle triangle if the other two sides are \(8\) and \(8\sqrt{3}\).

**Solution**

First, let's check the ratio to verify if it is suitable for a 30-60-90 triangle.

Ratio of the two sides = \(8 : 8\sqrt{3}\)

\(1 : \sqrt{3}\) indicates that the triangle is a 30-60-90 triangle.

We know that the hypotenuse is 2 times the smallest side.

Thus, the hypotenuse is \(2 \times 8 = 16\)

\(\therefore\) Hypotenuse = 16 |

Example 3 |

A triangle has sides \(2\sqrt{2}\), \(2\sqrt{6}\) and \(2\sqrt{8}\).

Find the angles of this triangle.

**Solution**

The sides of the triangle are \(2\sqrt{2}\), \(2\sqrt{6}\) and \(2\sqrt{8}\).

First, let's check whether the sides are following the 30-60-90 triangle rule or not.

\(2\sqrt{2} : 2\sqrt{6} : 2\sqrt{8}\) can be re-written as \(2\sqrt{2} : 2\sqrt{2} \times \sqrt{3} : 2 \times 2\sqrt{2}\)

Divide by \(2\sqrt{2}\)

We get,

\(1 : \sqrt{3} : 2\)

The sides are following the 30-60-90 triangle rule.

Hence, the angles of the triangle are \(30{^\circ}, 60{^\circ}, \text{ and } 90{^\circ}\)

\(\therefore\) \(30{^\circ}, 60{^\circ}, \text{ and } 90{^\circ}\) |

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about 30 60 90 triangle with the simulations and practice questions. Now, you will be able to easily solve problems on the area of 30 60 90 triangle, 30 60 90 triangle rules, 30 60 90 triangle sides, 30 60 90 triangle calculator, 30 60 90 triangle formula, 30 60 90 triangle ratios, and 30 60 90 triangle theorem**.**

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**Frequently Asked Questions (FAQs)**

## 1. What is the perimeter of a 30 60 90 triangle?

The perimeter of a 30 60 90 triangle with the smallest side equal to \(a\) is the sum of all the three sides.

The other two sides are \(a\sqrt{3}\) and \(2a\).

The perimeter of the triangle is \(a + a\sqrt{3} + 2a\)

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

= \(a\sqrt{3}(1 + \sqrt{3})\)

## 2. Are there any tips for remembering the 30 60 90 triangle rules?

Remember it as 1, 3, 2; it can resemble the ratio of the sides, all you need to remember is that the middle term is \(\sqrt{3}\)