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).
30 60 90 Triangle
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.
⇒ c2 = x2 + (x√3)2
⇒ c2 = x2 + (x√3) (x√3)
⇒ c2 = x2 + 3x2
⇒ c2 = 4x2
Squaring both sides, we get,
√c2 = √4x2
c = 2x
Properties
The angles are in the ratio 1: 2: 3, which are in arithmetic progression
The sides are in the ratio 1: √3: 2 (x: √3x: 2x)
The side opposite the 30° angle is the shorter side, denote by x
The side opposite the 60° angle is the longer side, denote by x√3
The side opposite the 90° angle is the hypotenuse, denote by 2x
All 30-60-90 triangles are similar triangles
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
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, AB2 = AD2 + BD2 x2 = AD2 +(x/2)2 x2 – (x/2)2 = AD2 AD2 = 3x2/4 =x√3/2 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:
30 60 90 Triangle Formula
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.
Find the value of y and z in the given diagram.