Triangles are one of the most basic shapes in geometry and are used in a wide variety of mathematical applications. One particularly important type of triangle is the 30-60-90 triangle, which has some unique properties that make it useful in a number of mathematical contexts.

**What is 30-60-90 Triangle?**

A 30-60-90 triangle is a special type of right triangle that has angles measuring 30 degrees, 60 degrees, and 90 degrees. This means that one of the angles is half of the right angle, while the other is one-third of the right angle. The 30-60-90 triangle is an isosceles triangle, which means that two of the sides have the same length.

The sides of a 30-60-90 triangle have a specific ratio that is useful in a variety of mathematical applications. This ratio is based on the length of the hypotenuse, which is the longest side of the triangle that is opposite the right angle. The other two sides are called the adjacent and opposite sides.

- The ratio of the sides in a 30-60-90 triangle is as follows:
- The side opposite the 30-degree angle is half the length of the hypotenuse.
- The side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle.
- The hypotenuse is twice the length of the side opposite the 30-degree angle.

This means that if the length of the hypotenuse is known, the lengths of the other two sides can be easily calculated using this ratio. For example, if the hypotenuse has a length of 6 units, the side opposite the 30-degree angle would be 3 units, and the side opposite the 60-degree angle would be 3√3 units.

Here, in the triangle ABC, **∠** C = 30°,**∠** A = 60°, and **∠** B = 90° and in the triangle PQK, **∠** P = 30°,**∠** K = 60°, and **∠** Q = 90°

**What is 30-60-90 Triangle Formula?**

The 30-60-90 formula is useful in a variety of mathematical contexts, including trigonometry, geometry, and physics. For example, it can be used to calculate the height of an object that is at a known distance from an observer or the distance between two objects that are at a known height.

The 30-60-90 formula is a rule in mathematics that applies to special right triangles whose angles are in the ratio of 1:2:√3. Here are some of its properties or specifications:

- The angles in a 30-60-90 triangle are 30 degrees, 60 degrees, and 90 degrees.
- The sides of a 30-60-90 triangle are in the ratio of 1:√3:2.
- The hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg.
- The longer leg of a 30-60-90 triangle is √3 times as long as the shorter leg.
- The area of a 30-60-90 triangle can be calculated by taking half the product of the two shorter sides.
- The perimeter of a 30-60-90 triangle is equal to the sum of the lengths of all three sides.

The 30-60-90 formula can be used to find the missing side lengths or angles of a triangle if at least one side length or angle is known.

In addition to its mathematical applications, the 30-60-90 triangle also has some interesting properties from a visual perspective. When you draw a 30-60-90 triangle inside a circle, the side opposite the 30-degree angle will be the diameter of the circle, while the side opposite the 60-degree angle will be the radius of the circle.

In conclusion, the 30-60-90 formula is an important mathematical tool that has a wide range of applications in geometry, trigonometry, and physics. Its simple ratio of sides makes it easy to use and remember, and its unique properties make it a fascinating object to study from a visual perspective.

**30-60-90 Triangle Theorem:**

The 30-60-90 Triangle Theorem states that

In a triangle with angles of 30, 60, and 90 degrees, the hypotenuse has a length that is twice the length of the shortest side. Additionally, the length of the other side is equal to the square root of three times the length of the shortest side.

**30-60-90 Triangle Proof:**

Suppose we have an equilateral triangle ABC where all three sides are of equal length ‘a’.

Afterwards, draw a perpendicular line from vertex A to side BC, which intersects at point D on the side BC of the equilateral triangle ABC. It is worth noting that in an equilateral triangle, the perpendicular line bisects the opposite side.

Triangle ABD and ADC are two right-angled triangles with angles of 30, 60, and 90 degrees. As both triangles are similar, we can apply the Pythagorean theorem to find the length of AD.

Using the Pythagorean theorem, we get:

(AB)² = (AD)² + (BD)²

a² = (AD)² + (a/2)²

a² – (a/2)² = (AD)²

3a²/4 = (AD)²

Taking the square root on both sides, we get:

AD = (a√3)/2

BD = a/2

AB = a

These side lengths follow the same ratio of a/2 : (a√3)/2 : a, which simplifies to 1 : √3 : 2.

Thus, by multiplying the ratio by 2 and dividing by ‘a’, we obtain:

(2a)/(2a) : (2a√3)/(2a) : (2a/a)

This simplifies to 1 : √3 : 2, which is known as the 30-60-90 triangle theorem.

**30-60-90 Triangle Rule:**

The 30-60-90 triangle rule allows us to determine the measure of any of the three sides of a 30-60-90 triangle by knowing the measure of at least one side. The following table illustrates how to use the 30-60-90 triangle rule to determine the sides of a 30-60-90 triangle:

**Area of a 30-60-90 Triangle**

The formula for calculating the area of a triangle is given by (1/2) × base × height. In a right-angled triangle, the height refers to the length of the perpendicular from the vertex of the right angle to the hypotenuse. As a result, the formula for calculating the area of a right-angled triangle can be written as (1/2) × base × perpendicular.

Now, let’s apply this formula to determine the area of a 30-60-90 triangle.

Assuming that the base BC of the triangle ABC is equal to ‘a’, and the hypotenuse of the triangle is denoted by AC, we can use the formula we learned earlier to find the length of the perpendicular AD.

Recall that the length of the hypotenuse AC is given by 2a, where ‘a’ represents the length of the shortest side of the triangle.

Hence, the length of the perpendicular AD is:

AD = (a√3)/2

To find the area of the 30-60-90 triangle, we can use the formula for the area of a triangle, which is given by:

Area = (1/2) × base × height

In this case, the base is ‘a’, and the height is AD, which is (a√3)/2.

Thus, the area of the 30-60-90 triangle when the base (side of the middle length) is ‘a’ is:

Area = (1/2) × a × (a√3)/2

Area = a²/(2√3)

Therefore, the area of the 30-60-90 triangle when the length of the base is given as ‘a’ is equal to a squared divided by two times the square root of three (a²/(2√3)).

**Example 1**

A ladder is leaning against a wall. The base of the ladder is 6 feet away from the wall, and the top of the ladder is 8 feet away from the ground. What is the length of the ladder?

**Solution: **

We can model the situation as a right triangle, where the wall and the ground form the legs of the triangle, and the ladder forms the hypotenuse.

Let c be the length of the ladder. Then we have:

a = 6 (the distance from the wall to the base of the ladder) b = 8 (the height of the ladder from the ground)

Using the Pythagorean theorem, we have:

c^2 = a^2 + b^2 c^2 = 6^2 + 8^2 c^2 = 36 + 64 c^2 = 100 c = 10

Therefore, the length of the ladder is 10 feet.

**Example 2**

We are given a triangle with sides 2√2, 2√6, and 2√8. We want to find the angles of this triangle.

**Solution:**

We can use the law of cosines to find the angles of the triangle. The law of cosines states that in a triangle with sides a, b, and c, the cosine of one of the angles, say A, is given by:

cos(A) = (b^2 + c^2 – a^2) / (2bc)

Similarly, we can find the cosines of the other two angles, say B and C, by using the same formula with the appropriate sides.

Using this formula, we have:

cos(A) = (2√6)^2 + (2√8)^2 – (2√2)^2 / (2 * 2√6 * 2√8) cos(A) = 24 + 32 – 8 / (4√12) cos(A) = 48 / (4√12) cos(A) = 4√3 / 9

Similarly, we can find the cosines of angles B and C:

cos(B) = (2√2)^2 + (2√8)^2 – (2√6)^2 / (2 * 2√2 * 2√8) cos(B) = 8 + 32 – 24 / (4√16) cos(B) = 16 / (4 * 2) cos(B) = 1/2

cos(C) = (2√2)^2 + (2√6)^2 – (2√8)^2 / (2 * 2√2 * 2√6) cos(C) = 8 + 24 – 32 / (4√12) cos(C) = 0

Now that we have the cosines of the three angles, we can use the inverse cosine function to find the angles themselves:

A = cos^-1(4√3 / 9) ≈ 38.24° B = cos^-1(1/2) = 60° C = cos^-1(0) = 90°

Therefore, the angles of the triangle are approximately 38.24°, 60°, and 90°.

**Example 3**

We are given a triangle with sides of 4 units, √48 units, and 8 units. We want to verify whether this triangle is a 30-60-90 triangle.

Recall that a 30-60-90 triangle is a right triangle with angles of 30°, 60°, and 90°, and with side lengths in the ratio 1:√3:2.

**Solution:**

To verify whether the given triangle is a 30-60-90 triangle, we need to check whether its side lengths are in the ratio 1:√3:2.

Let’s start by dividing the longest side, 8 units, by the medium side, √48 units:

8 / √48 = 8 / (4√3) = 2 / √3

Now let’s divide the medium side, √48 units, by the shortest side, 4 units:

√48 / 4 = √(16*3) / 4 = 2√3 / 2 = √3

So the ratio of the side lengths is:

4 : √48 : 8 = 1 : √3 : 2

Since the ratio of the side lengths is in form 1:√3:2, which is the ratio for a 30-60-90 triangle, we can conclude that the given triangle is indeed a 30-60-90 triangle.

Therefore, the angles of the triangle are 30°, 60°, and 90°, and the side lengths are in the ratio 1:√3:2.