# 30-60-90 triangle: definition, formula, theorem &amp; solved examples (2023)

## What is the 30-60-90 triangle?

It is a triangle with the same angles of 30, 60 and 90. This triangle is always a right triangle because one of the angles is 90 degrees. As mentioned, it is a unique triangle with unique length and angle values.

The 30-60-90 triangle is called a quirky right triangle because its angles have a unique 1:2:3 ratio. A right triangle is any triangle that has an angle of 90°. A 30-60-90 triangle is a definite right triangle with angles of 30°, 60° and 90° at all times. Here are some variations of the 30-60-90 triangle.

30-60-90 triangle

Also read: Uneven, acute and obtuse triangles

## sides of the 30-60-90 triangle

The side lengths of a 30-60-90 triangle are constantly in continuous connection with each other, making it a unique triangle.

∠N = 30°, ∠R = 60° and ∠M = 90° in the 30-60-90 triangle ABC shown below. The following definitions will help us understand the connection between the two sides:

• Since 30° is the smallest angle in this triangle, the side opposite the 30° angle, RM = x, will always be the smallest.
• Since 60° is the middle degree angle in this triangle, the side opposite the 60° angle MN = x × √3 = x√3 will have the middle length.
• Since 90° is the largest angle, the hypotenuse RN = 2x is the largest side on the opposite side of the 90° angle.

The basic aspect ratio of the triangle is 30-60-90:

 The side that is on the other side of the 30° angle. X The side that is on the other side of the 60° angle. x * √3 The side that is on the other side of the 90° angle. 2x

The sides of a 30-60-90 triangle are always in a 1:√3:2 ratio in a 30-60-90 triangle. For sides, this is also known as the 30-60-90 triangle formula. x:x√3:2x. In the 30-60-90 triangle proof section we will discover how to derive this ratio.

30-60-90 triangle formula

## 30-60-90 Theorem of Triangles Proof

Consider the equilateral triangle ABC, which has side length 'a'.

Now, at point D of triangle ABC, draw a perpendicular from apex A to side BC. In an equilateral triangle, the perpendicular bisects the other side.

ABD Triangle & ADC consists of two 30-60-90 triangles. Both triangles are right triangles and comparable. As a result, we can usePythagorean theoremto calculate the length of AD.

BD = a/2

(Video) 30-60-90 Triangles(HD)

AB = a

These sides also have the same ratio a/2: (a3√3)/2: a.

Divide by 'a' and multiply by 2.

(2a)/(2a) : (2a√3)/(2a): (2a/a)

1:√3:2 is the result. It's called the 30-60-90 triangle theorem.

Examples of 30-60-90 triangles

Consider the following cases of a triangle with side lengths of 30-60-90 degrees:

Example 1:

DEF is a 30-60-90 triangle.

Wo,

∠F = 30°, ∠D = 60° and ∠E = 90°

• DE = y = 2 is the side opposite the 30° angle.
• BC = y√3 = 2√3 is the side opposite the 60° angle.
• The hypotenuse AC = 2y = 2x2 = 4 is the side opposite the 90° angle.

Example 2:

PQR is the 30-60-90 triangle.

Wo,

∠ R = 30 degrees, ∠ P = 60 degrees and ∠ Q = 90 degrees

• AB = y = 7 is the side opposite the 30° angle.
• BC = y√3 = 7√3 is the side opposite the 60° angle.
• The hypotenuse AC = 2y = 2 x7 = 14 is the side opposite the 90° angle.

## 30-60-90 triangle rule

The measure of each of the three sides of a 30-60-90 triangle can be determined by knowing the measure of at least one of the sides of the triangle. It's called the 30-60-90 triangle rule. The following table explains how to use the 30-60-90 triangle rule to get the sides of a 30-60-90 triangle:

• When the base is given

The base BC of the triangle is assumed to be 'a'.

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

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

• If the perpendicular is given

The base BC of the triangle is assumed to be 'a'.

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

(Video) 30-60-90 Special Right Triangles

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

• Given the hypotenuse

The base BC of the triangle is assumed to be 'a'.

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

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

Cosec cot formula

## Area of ​​the 30-60-90 triangle

The area of ​​a triangle is calculated using the formula = (1/2)*base*height. The height of a right triangle is the perpendicular of the triangle. As a result, the area of ​​a right triangle is calculated using the formula = (1/2) * base * perpendicular.

Let's look at how to use this formula to calculate the area of ​​a triangle with sides of 30-60-90 degrees.

The base BC of the triangle is said to be 'a' while the hypotenuse ABC of the triangle is said to be AC. We learned in the previous section how to find the hypotenuse given the base.

Let's apply everything we've learned about the formula.

As a result, the perpendicular of the triangle is equal to a /√ 3

Area of ​​the triangle = 1/ 2 × a × a/ √ 3

As a result, the area of ​​the 30-60-90 triangle when the base is 'a is: a2/(2√3)

Also read: area of ​​a triangle

## things to remember

• A 30-60-90 triangle, pronounced "30-60-90," is a particularly unique shape of a triangle.
• Because the angles of the 30-60-90 triangle are in a unique 1:2:3 ratio and the sides are in a 1:√3:2 ratio, this triangle is known as a special right triangle.
• A 30-60-90 triangle is a unique right triangle with 30°, 60°, and 90° angles.
• If only one side of a 30-60-90 triangle is known, all of its sides can be determined. It's called the 30-60-90 triangle rule.
• Since 30 degrees is the shortest angle, the opposite side is always smaller. Since 60 degrees is the middle degree angle in this triangle, the side opposite the 60° angle is the middle longitude. Finally, since 90 degrees is the largest angle, the side opposite 90 degrees is always the largest side (the hypotenuse).

## sample questions

queue If the other two sides of a right triangle are 8 and 8√ 3 units, find the length of the hypotenuse.(3 Mark)

Ans.Let's start by checking the ratio to see if it works for a 30-60-90 triangle.

8:8√3 ⇒ 1:√3 is the ratio of the two sides.

So the triangle is a 30-60-90 triangle. The hypotenuse is known to be 2 times the smallest side.

As a result, the hypotenuse is 2*8 = 16.

Hypotenuse = 16 units is the answer.

queue The diagonal of a right triangle is 8 cm. Since one of the angles of the triangle is 30 degrees, you get the lengths of the other two sides.(3 Mark)

Ans.This triangle must be 30°-60°-90°. As a result we use the x: x√3:2x..

8 cm = Diagonale = Hypotenuse

2x corresponds to 8 cm

x = 4cm

Ersatz.

x√3 = 4√3cm

The right triangle has a shorter side of 4 cm and a longer side of 4√3 cm.

(Video) 30-60-90 triangle side ratios proof | Right triangles and trigonometry | Geometry | Khan Academy

queue The two sides of a triangle are 5√3mm and 5mm long. Find the length of the hypotenuse.(3 Mark)

Ans.Check the aspect ratio to see if it matches the ratio x:x√3:2x.

5: 5√3:x = 1(5): √3 (5):x

As a result, x = 5

Multiply 2 by 5 to get the answer.

Multiplying 2x by 5 equals ten.

As a result, the hypotenuse is 10 mm.

queue Find the hypotenuse of a 30°-60°-90° triangle with a side 6 inches longer.(3 Mark)

Ans.Ratio = x: x√3:2x.

⇒ x√3 = 6 inches.

Square both sides

⇒ (x√3)2= 36

⇒ 3x2= 36

⇒ X2= 12

⇒ x =√12

queue The sides of a triangle are 2√2, 2√6, and 2√8. Find the angles of the triangle.(3 Mark)

Ans.The triangle has sides of 2√2, 2√6, and 2√8.

(Video) 30-60-90 triangle example problem | Right triangles and trigonometry | Geometry | Khan Academy

First, let's look at the sides to see if they follow the 30-60-90 triangle rule.

2√2:2√6:2√8 can be rewritten as 2√2:2√2×√3:2×2√2

We get 1:√3:2 by dividing the ratio by 2√2.

On these pages, the triangle rule is 30-60-90.

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

queue Find the missing side of the triangle.(3 Mark)

Ans.Since this is a right triangle, the hypotenuse is the same length as one of the sides of the triangle. Hence it is known as a 30-60-90 triangle, with a smaller angle of 30. In the image shown, the longer side is always opposite 60° and the missing side is 3√3 units.

queue What is the formula for the triangle 30 60 90?(2 Mark)

Ans.The sides of a 30-60-90 triangle are always in a 1:√3:2 ratio in a 30-60-90 triangle. For sides, this is also known as the 30-60-90 triangle formula. x:x√3:2x.

queue What is the application of the 30 60 90 triangle theorem?(3 Mark)

Ans.It turns out that you can find the measurement of each of the three sides of a 30-60-90 triangle by knowing the measurement of at least one of the triangle's sides. The hypotenuse is the side opposite the 30 degree angle that is twice the length of the leg that is shorter than the other.

queue How do you solve a right triangle with a hypotenuse of 30 60 90 when you only have the hypotenuse?(2 Mark)

Ans.In any 30-60-90 triangle you can observe the following: the shortest leg is opposite the 30 degree angle, the hypotenuse is always twice the length of the shortest leg, and the length of the long leg can be found by multiplying the short leg be leg divided by the square root of 3.

queue How many triangles can have angles of 60, 90, and 30?(1 Mark)

Ans.Only one triangle with angles 60, 90 and 30 may be drawn. because the measure of three angles is equal to 180 due to the angle-sum property.

queue What are the lengths of the two remaining sides of a right triangle if one of the angles is 30 degrees and the shortest side is 7 m?(3 Mark)

Ans.The lengths of the sides of this triangle are in the ratio x:x√3:2x, i.e. a 30-60-90 triangle.

For the longer leg and hypotenuse, use x = 7m.

⇒ x √3 = 7√3

⇒ 2x = 2(7) =14

As a result, the other sides are 14 and 7√3 meters long.

queue When to use 30-60-90 triangles?(5 Mark)

Ans.For geometry problems, we can easily answer missing information like angular measures and side lengths by knowing three pieces of information, one of which is that the triangle is a right triangle. It is possible to fill in the rest of the triangle with three pieces of information, usually two protractors and one side, or one protractor and two sides.

When struggling with trigonometry, understanding the basic concepts of sine, cosine, and tangent makes it fairly easy to find the values ​​for these in any 30-60-90 triangle. The terms sine, cosine, and tangent all refer to a ratio of the sides of a triangle based on one of the angles, theta or. Because the sides of any 30-60-90 triangle have the same ratio, the values ​​for sine, cosine, and tangent are always the same, particularly the following two, which are commonly used in standardized tests:

• Sine 30 equals 1/2
• cosine60 equals 1/2.

Check more:

(Video) 30-60-90 Triangle Theorem - Proof | Don't Memorise

 Isosceles right triangle Introduction to trigonometry Properties of the triangle trigonometry values Sin-Cos-Formeln Analytic Geometry Sin 30 degrees value of cos 60 trigonometric ratios Trigonometry Trigonometry Tables Trigonometric Identities

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