306090 Triangle
There are different types of triangles such as obtuse, isosceles, acute, equilateral, and so on. But only a few types of triangles are considered special triangles. These triangles are special as their sides and angles are consistent and predictable. Their properties can be used to solve various geometry or trigonometry problems. A 306090 triangle—pronounced "thirty sixty ninety"—is one such very special type of triangle indeed.
In this lesson, we will learn all about the 306090 triangle, its definition, sides, area, and the rules that apply to these triangles.
306090 Triangle
The 306090 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3. Here, a right triangle means being any triangle that contains a 90° angle. A 306090 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°. Here are some of the variants of a 306090 triangle. The triangles ABC and PQK are 306090 triangles.
Here, in the triangle ABC, ∠ C = 30°,∠ A = 60°, and ∠ B = 90° and in the triangle PQK, ∠ P = 30°,∠ K = 60°, and ∠ Q = 90°
Sides of a 306090 Triangle
A 306090 triangle is a special triangle since the length of its sides is always in a consistent relationship with one another.
In the belowgiven 306090 triangle ABC, ∠ C = 30°,∠ A = 60°, and ∠ B = 90°. We can understand the relationship between each of the sides from the below definitions:

The side that is opposite to the 30° angle, AB = y will always be the smallest since 30° is the smallest angle in this triangle

The side that is opposite to the 60° angle, BC = y × √3 = y√3 will be the medium length because 60° is the midsized degree angle in this triangle

On the side that is opposite to the 90° angle, the hypotenuse AC = 2y will be the largest side because 90° is the largest angle.
In a 306090 triangle, the ratio of the sides is always in the ratio of 1:√3: 2. This is also known as the 306090 triangle formula for sides. y:y√3:2y. Let us learn the derivation of this ratio in the 306090 triangle proof section.
Consider some of the examples of a 306090 degree triangle with these side lengths:
Here, in the 306090 triangle DEF ∠ F = 30°,∠ D = 60°, and ∠ E = 90°

Here, in the 306090 triangle PQR ∠ R = 30°,∠ P = 60°, and ∠ Q = 90°

306090Triangle Theorem 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 306090 triangles. Both the triangles are similar and rightangled 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 306090 triangle theorem.
306090 Triangle Rule
In a 306090 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 306090 triangle rule. The belowgiven table shows how to find the sides of a 306090 triangle using the 306090 triangle rule:
Base is given  Perpendicular is given  Hypotenuse is given 

The Base BC of the triangle is assumed to be 'a'. 
The perpendicular DE of the triangle is assumed to be 'a'. 
The hypotenuse PR of the triangle is assumed to be 'a'. 
The perpendicular of the triangle ABC is AB = (a /√3) The hypotenuse of the triangle ABC is AC = (2a)/√3 
The base of the triangle DEF is EF = √3a. The hypotenuse of the triangle DEF is DF = 2a. 
The base of the triangle PQR is QR = (√3a)/2. The perpendicular of the triangle PQR is PQ = (a/2). 
Area of a 306090 Triangle
The formula to calculate the area of a triangle is = (1/2) × base × height. In a rightangled triangle, the height is the perpendicular of the triangle. Thus, the formula to calculate the area of a rightangle triangle is = (1/2) × base × perpendicular
Let's learn how to apply this formula to find the area of the 306090 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 = \(\frac{a}{\sqrt{3}}\)
Area of the triangle = (1/2) × base × perpendicular
Area = \(\frac{1}{2} \times a \times \frac{a}{\sqrt{3}}\)
Therefore, the area of the 306090 triangle when the base is given as 'a' is: a^{2}/(2√3)
Related Topics
Check out these interesting articles to learn more about the 306090 triangle and its related topics
Important Notes
Here is a list of a few points that should be remembered while studying 306090 triangles: The 306090 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 306090 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°
 All the sides of a 306090 triangle can be calculated if any one side is given. This is called the 306090 triangle rule.
Solved Examples

Example 1: Find the length of the hypotenuse of a rightangle 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 306090 triangle.
The ratio of the two sides = 8:8√3 ⇒ 1:√3
This indicates that the triangle is a 306090 triangle. We know that the hypotenuse is 2 times the smallest side.
Thus, the hypotenuse is 2 × 8 = 16
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 306090 triangle rule.
2√2: 2√6: 2√8 can be rewritten as 2√2: 2√2 × √3: 2 × 2√2
If we divide the ratio by 2√2, we get 1:√3: 2
These sides are following the 306090 triangle rule.
Answer: The angles of the triangle are 30°,60°, and 90°
FAQs on 306090 Triangle
What Is the Perimeter of a 306090 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 306090 Triangle Rules?
This method can be used to remember the 306090 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 Is a 306090 Triangle?
The 306090 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3. A 306090 triangle is a special right triangle that always has angles of measure 30°, 60°, and 90°.
What Are the Side Lengths of a 306090 Triangle?
The sides of a 306090 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 midsized 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 454590 Triangle?
A 454590 triangle has a right angle and two 45 degree angles. The two sides of a 454590 triangle are always equal and the hypotenuse of the triangle is always opposite to the right angle.
What Are Some Similarities Between 306090 Triangles and 454590 Triangles?
These are some of the similarities between 306090 triangle and 454590 triangle: both are not acute triangles, both are rightangle 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°