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There are many types of triangles. Acute, Obtuse, Isosceles, Equilateral triangles. Out of these different triangles, few of them are special. When we say special, it means the sides and angles which are predictable and consistent.

**Thirty sixty ninety triangle**

Out of all the other shortcuts, 30-60-90 is indeed a special Triangle.

## What is a 30-60-90 Triangle?

It is a triangle where the angles are always 30, 60 and 90. As one angle is 90, so this triangle is always a right triangle. Thus, these angles form a right-angled triangle. Also, the sum of two acute angles is equal to the right angle, and these angles will be in the ratio 1 : 2 or 2 : 1.

## Sides of a 30 60 90 Triangle

As explained above, it is a unique triangle with particular values of lengths and angles. Thus, the sides of 30 60 and 90 triangles are considered to be the Pythagorean triples. In general, the sides of a triangle with angles 30 degrees, 60 degrees and 90 degrees can be expressed as given in the below table:

The basic 30-60-90 triangle sides ratio is: | |

The side opposite the 30° angle | x |

The side opposite the 60° angle | x * √3 |

The side opposite the 90° angle | 2x |

**Facts about the sides of 30 60 90 triangle:**

**The side opposite to the angle 30° is always the shortest since 30 degrees is the smallest angleThe side opposite to the angle 60° will be the medium length since 60 degrees is the mid-sized degree angle in this triangleThe side opposite to the angle 90° will always be the largest since 90 degrees is the largest. This side of the triangle is called the hypotenuse**

**Area of 30 60 90 Triangle FormulaConsider the triangle of 30 60 90 in which the sides can be expressed as:**

Here,

Base = x√3

Perpendicular (or Height) = x

Hypotenuse = 2x

We know that,

Area of triangle = (½) × Base × Height

= (½) × (x√3) × (x)

= (√3/2)x^{2}

## Example of 30 – 60 -90 rule

**Example: **Find the missing side of the given triangle.

**Solution:**

As it is a right triangle in which the hypotenuse is the double of one of the sides of the triangle. Thus, it is called a 30-60-90 triangle where a smaller angle will be 30. The longer side is always opposite to 60° and the missing side measures 3√3 units in the given figure.

Consider some of the examples of a 30-60-90 degree triangle with these side lengths:

## 30-60-90-Triangle Theorem

The statement of the 30-60-90-Triangle Theorem is given as,

**Statement: **The length of the hypotenuse is twice the length of the shortest side and the length of the other side is √3 times the length of the shortest side in a 30-60-90-Triangle.

## 30-60-90-Triangle Proof

Let’s consider an equilateral triangle ABC with a 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.

(AB)^{2 }= (AD)^{2} + (BD)^{2}

a^{2} = (AD)^{2} + (a/2)^{2}

a^{2} – (a/2)^{2} = (AD)^{2}

3a^{2}/4 = (AD)^{2}

(a**√**3)/2 = AD

AD = (a**√**3)/2

BD = a/2

AB = a

These sides also follow the same ratio a/2 : (a√3)/2: a

Multiply by 2 and divide by ‘a’,

(2a)/(2a) : (2a√3)/(2a): (2a/a)

We get 1:√3:2. This is the 30-60-90 triangle theorem.

## 30-60-90 Triangle Rule

In a 30-60-90 triangle, the measure of any of the three sides can be found out by knowing the measure of at least one side in the triangle. This is called the 30-60-90 triangle rule. The below-given table shows how to find the sides of a 30-60-90 triangle using the 30-60-90 triangle rule:

The perpendicular of the triangle ABC is AB = (a /√3)

The hypotenuse of the triangle ABC is AC = (2a)/√3

The perpendicular DE of the triangle is assumed to be ‘a’.

The base of the triangle DEF is EF = √3a.

The hypotenuse of the triangle DEF is DF = 2a.

The hypotenuse PR of the triangle is assumed to be ‘a’.

The base of the triangle PQR is QR = (√3a)/2.

The perpendicular of the triangle PQR is PQ = (a/2).

## Area of a 30-60-90 Triangle

The formula to calculate the area of a triangle is = (1/2) × base × height. In a right-angled triangle, the height is the perpendicular of the triangle. Thus, the formula to calculate the area of a right-angle triangle is = (1/2) × base × perpendicular

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

Base BC of the triangle is assumed to be ‘a’, and the hypotenuse of the triangle ABC is AC. We have learned in the previous section how to find the hypotenuse when the base is given.

Let’s apply the formula we have learned.

Thus, perpendicular of the triangle = a/√3

Area of the triangle = (1/2) × base × perpendicular

Area = 1/2 × a × a/√3

Therefore, the area of the 30-60-90 triangle when the base (side of middle length) is given as ‘a’ is: a^{2}/(2√3)

**Important Notes on 30-60-90 Triangle**

Here is a list of a few points that should be remembered while studying 30-60-90 triangles:

- The 30-60-90 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3 and the sides are in the ratio 1:√3: 2
- A 30-60-90 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°
- All the sides of a 30-60-90 triangle can be calculated if any one side is given. This is called the 30-60-90 triangle rule.

## Solved Examples on 30-60-90 Triangle

**Example 1:** Find the length of the hypotenuse of a right-angle triangle if the other two sides are 8 and 8**√** 3 units.

**Solution:**First, let’s check the ratio to verify if it is suitable for a 30-60-90 triangle.The ratio of the two sides = 8:8√3 = 1:√3This 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 × 8 = 16 units

**Answer: **Hypotenuse = 16 units

**Example 2:** A triangle has sides 2√2, 2√6, and 2√8. Find the angles of this triangle.

**Solution:**The sides of the triangle are 2√2, 2√6, and 2√8.First, let’s check whether the sides are following the 30-60-90 triangle rule.2√2: 2√6: 2√8 can be re-written as 2√2: 2√2 × √3: 2 × 2√2If we divide the ratio by 2√2, we get 1:√3: 2These sides are following the 30-60-90 triangle rule. We know that the angles of a 30-60-90 triangle are 30°,60°, and 90°.

**Answer:** The angles of the triangle are 30°,60°, and 90°.

**Example 3:** Verify if the triangle is a 30-60-90 triangle if the sides of the triangle are 4 units, √48 units, and 8 units.

**Solution: **To verify if the triangle is a 30-60-90 triangle, we will check the ratio of the sides. We have4 : √48 units : 8 = 4 : 4√3 : 4 × 2= 1 : √3 : 2Using the 30-60-90 triangle theorem, the sides are in the ratio 1 : √3 : 2. So the triangle is a 30-60-90 triangle.

**Answer:** The given triangle is a 30-60-90 triangle.

## FAQs on 30-60-90 Triangle

**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 three sides. The other two sides are a√3 and 2a. The perimeter of the triangle is a+a√3+2a = 3a+a√3 = a√3(1+√3).

**Are There Any Tips for Remembering the 30-60-90 Triangle Rules?**

This method can be used to remember the 30-60-90 triangle rule. One can remember it as 1, 3, 2; it can resemble the ratio of the sides, all one needs to remember is that the middle term is √3

**What Are the Side Lengths of a 30-60-90 Triangle?**

The sides of a 30-60-90 triangle have a set pattern. The side that is opposite to the 30° angle, ‘y’ will always be the smallest since 30° is the smallest angle in this triangle. The side that is opposite to the 60° angle, y√3 will be the medium length because 60° is the mid-sized degree angle in this triangle. The side that is opposite to the 90° angle, 2y will be the largest side because 90° is the largest angle.

**What Are the Rules for a 45-45-90 Triangle?**

A 45-45-90 triangle has a right angle and two 45 degree angles. The two sides of a 45-45-90 triangle are always equal and the hypotenuse of the triangle is always opposite to the right angle.

**What Are Some Similarities Between 30-60-90 Triangles and 45-45-90 Triangles?**

These are some of the similarities between 30-60-90 triangle and 45-45-90 triangle: both are not acute triangles, both are right-angle triangles, both are not obtuse triangles, the square of the hypotenuse equals the sum of the squares of the other two sides for both triangles, and the sum of the interior angles of both are 180°.

**Which Leg is the Long Leg in the 30-60-90 Triangle?**

The long leg of a 30-60-90 Triangle is the leg whose length is greater than the shortest leg and ess than the hypotenuse. The length of the long leg is equal to √3 times the length of the shortest leg.

**30 60 90 Triangle**

**30-60-90 Triangle Theorem – Proof**

**30-60-90 Special Right Triangles**

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