# 30-60-90 Triangle

A 30-60-90 triangle is a special right triangle whose three angles are 30°, 60°, and 90°. The triangle is special because its side lengths are in the ratio of 1: √3: 2 (x: x√3: 2x for shorter side: longer side: hypotenuse).

Since a 30-60-90 triangle is a right triangle, the Pythagoras formula a2 + b2 = c2, where a = longer side, b = shorter side, and c = hypotenuse is also applicable. For example, the hypotenuse can be obtained when the two other sides are known as shown below.

⇒ c= x+ (x√3)2

⇒ c= x+ (x√3) (x√3)

⇒ c2 = x+ 3x2

⇒ c2 = 4x2

Squaring both sides, we get,

√c= √4x2

c = 2x

## Properties

1. The angles are in the ratio 1: 2: 3, which are in arithmetic progression
2. The sides are in the ratio 1: √3: 2 (x: √3x: 2x)
3. The side opposite the 30° angle is the shorter side, denote by x
4. The side opposite the 60° angle is the longer side, denote by x√3
5. The side opposite the 90° angle is the hypotenuse, denote by 2x
6. All 30-60-90 triangles are similar triangles
7. Two 30-60-90 triangles sharing the longer side form an equilateral triangle

## Rules

From the above properties, we get some basic rules applicable in all 30-60-90 triangles. The three side lengths are always in the ratio of 1: √3: 2 and the shortest side is always the smallest angle (30°), while the longest side is always opposite the largest angles (90°). These rules are useful for solving the 30-60-90 theorem that we will deal with in the next section.

## 30-60-90 Triangle Theorem

Thus, the properties 2, 3, 4, and 5 are collectively called the 30-60-90 triangle theorem, which is summarized below:

• The hypotenuse is twice the length of the short leg
• The length of the longer side is √3 times the shorter side

### 30-60-90 Triangle Theorem Proof To prove 30-60-90 Triangle Theorem

To prove:

Let △ABC be an equilateral triangle with each side length equal to x.  Proof:

Given: △ABC is an equilateral triangle with side length ‘x’
Proof:
A perpendicular line is drawn from vertex A to side BC that meets at point D such that it bisects the side BC. The two triangles formed, △ABD and △ADC are similar.
Since, both the triangles are right triangles; here we will use the Pythagorean Theorem to find the length of AD.
Thus,
BD = x/2
AB = x
Thus, the three sides are in the ratio of:
x/2: x√3/2: x
Multiplying by 2, we get,
1: √3: 2
Hence proved that the given △ABC is a 30-60-90 Triangle.

## How to Solve a 30-60-90 Triangle

Given the length of one side of a triangle, we can find the other side(s) without using long-step methods such as Pythagorean Theorem and trigonometric functions.

Solving a 30-60-90 triangle can have four possibilities:

• Possibility 1: When the shorter side is known, we can find the longer side by multiplying the shorter side by √3. The hypotenuse can then be obtained by Pythagoras Theorem.

Thus,

Longer side = shorter side × √3

• Possibility 2: When the longer side is known, we can find the shorter side by dividing the longer side by √3. The hypotenuse can then be obtained by Pythagoras Theorem.

Thus,

Shorter side = longer side/√3

• Possibility 3: When the shorter side is known, we can find the hypotenuse by multiplying the shorter side by 2. The longer side can then be obtained by Pythagoras Theorem.

Thus,

Hypotenuse = shorter side × 2

• Possibility 4: When the hypotenuse is known, we can find the shorter side by dividing the hypotenuse by 2. The longer side can then be obtained by Pythagoras Theorem.

Thus,

Shorter side = hypotenuse/2

## Formulas

The formulas of a 30-60-90 triangle when the length of the shorter side is x units are given below:

## How to Find a 30-60-90 Triangle

Let us solve some examples to understand the concepts better.

A right triangle whose one angle is 60 degrees has the longer side of 12√3 cm. Find the length of its shorter side and the hypotenuse.

Solution:

Since, the given triangle is a 30-60-90 triangle,
Its side lengths = x: x√3: 2x, here x√3 = longer side = 12√3 cm
Thus,
x√3 =12√3
Squaring both sides we get,
⇒ (x√3)2 = (12√3)2 cm
⇒ 3x2 = 144 x 3
⇒ x2 = 144
⇒ x = 12 cm
Hypotenuse = 2x = 2 x 12 = 24 cm Hence the shorter side is 12cm and hypotenuse is 24 cm.

A right triangle whose one angle is 60 degrees has the longer side of 12√3 cm. Find the length of its shorter side and the hypotenuse.

Solution:

Since, the given triangle is a 30-60-90 triangle,
Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here x√3 = longer side = 12√3 cm
Thus,
x√3 =12√3
Squaring both sides we get,
⇒ (x√3)2 = (12√3)2 cm
⇒ 3x2 = 144 x 3
⇒ x2 = 144
⇒ x = 12 cm
Hypotenuse = 2x = 2 x 12 = 24 cm Hence the shorter side is 12cm and hypotenuse is 24 cm. The diagonal of a right triangle is 14 cm, find the lengths of the other two sides of the triangle given that one of its angles is 30 degrees.

Solution:

Since, the given triangle is a 30-60-90 triangle,
Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here 2x = diagonal = hypotenuse = 12√3 cm
Thus,
⇒ 2x = 14 cm
⇒ x = 7 cm
Substituting the value of x, we get,
Longer side = x√3 = 7√3 cm
Hence, the length of the longer side is 7√3 cm. Find the value of y and z in the given diagram. Solution:

Since, the given triangle is a 30-60-90 triangle,
The side measuring 18m is the shorter side because it is opposite the 30-degree angle.
Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here x = 18m
Substituting the value of x, we get,
Given,
x√3 = y
Longer side = y = 18√3m
Given,
2x = z
Hypotenuse = z = 2 x 18 = 36m
Hence, the length of the hypotenuse is 36m.

If one angle of a right triangle is 30° and the measure of the shortest side is 11cm. Find the measure of the remaining two sides.

Solution:

Since, the given triangle is a 30-60-90 triangle,
Ratios of a 30-60-90 triangle side lengths = x: x√3: 2x, here x = 7cm
Substituting the value of x, we get,
Longer side = x√3 = 7√3cm
Hypotenuse = 2x = 2 x 7 = 14cm
Hence, the length of the other two sides is 7√3cm and 14cm.

A ramp making an angle of 30 degrees with the ground is used to offload a lorry that is 8 m high. Calculate the length of the ramp.

Solution:

Since, the ramp makes a 30-60-90 triangle with the ground,
Here,
Shorter side = x = 8m
Since, the length of the ramp is the hypotenuse of the given 30-60-90 triangle,
2x = 2 x 8 = 16m
Hence the length of the ramp is 16m.