30 60 90 Triangle – Definition with Examples
A triangle is generally known as a three-sided polygon. This means that for a shape to be classified as a triangle, it must possess 3 sides, 3 vertices, and 3 angles. Based on this definition, there are several categories of triangles: acute, obtuse, isosceles, equilateral, or scalene. There are however some triangles that do not necessarily follow the general description of what a triangle is due to some distinctive properties. These triangles are called special triangles.
This article discusses the special right triangles 30 60 90. Read on to find out what they are, which rules govern their use, and the related theorem.
What Is a 30-60-90 Triangle?
Simply put, 30-60-90 triangles are triangles whose angles are invariably 30, 60, and 90. Since one of its angles is 90, a 60 30 90 triangle classifies as a special right triangle and thus forms a right-angled triangle.
As one angle is 90, this triangle is always a right one. Also, the sum of two acute angles is equal to the right angle, and these angles will be in the ratio of 1: 2 or 2: 1.
A special right triangle with angles 30°, 60°, and 90° is called a 30-60-90 triangle. The angles of a 30-60-90 triangle are in the ratio 1: 2 : 3. Since 30° is the smallest angle in the triangle, the side opposite to the 30 degree triangle is always the smallest (shortest leg). The side opposite to the 60° angle is the longer leg, and finally, the side opposite to the 90° angle is the largest side of the right triangle, also known as the hypotenuse.
Since the 30-60-90 triangle is a special triangle, the side lengths of the 30-60-90 triangle are in a constant relationship.
30 60 90 Triangle Sides
The 30-60-90 triangle sides are in a specific ratio to each other. The side opposite to the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is the product of the hypotenuse and the square root of three divided by two. Finally, the hypotenuse is twice the length of the side opposite the 30-degree angle.
In other words, if we let x be the length of the side opposite the 30-degree angle, then the length of the hypotenuse is 2x, and the length of the side opposite the 60-degree angle is x√3.
This relationship between the 30 60 90 triangle side lengths is the key to solving geometry and trigonometry problems involving these types of triangles. Knowing the length of one side allows you to easily find the lengths of the other sides, which can be useful in various applications.
30 60 90 Triangle Rules
The 30-60-90 triangle has some unique properties that make it different from other triangles. The most important rule for a 30-60-90 triangle is that the length of the hypotenuse is always twice the length of the shorter leg. In other words, if the length of one leg is x, then the length of the hypotenuse is 2x. The longer leg, which is opposite to the 60-degree angle, is always √3 times the length of the shorter leg. That is, if the shorter leg is x, then the longer leg is √3x.
Here are some other important 30-60-90 rules:
- The sum of the angles in a 30-60-90 triangle is always 180 degrees.
- The ratio of the sides in a 30-60-90 triangle is 1:√3:2. That is, the ratio of the length of the shorter leg to the longer leg is √3:1, and the ratio of the length of the shorter leg to the hypotenuse is 1:2.
- The height of an equilateral triangle is √3 times the length of the shorter side, which is also the shorter leg of a 30-60-90 triangle.
These rules can be used to solve a variety of problems involving 30-60-90 triangles. For example, if you know the length of one leg of a 30-60-90 triangle, you can easily find the length of the other two sides using the ratios mentioned above. Similarly, if you know the length of the hypotenuse, you can find the lengths of the other two sides using the fact that the hypotenuse is twice the length of the shorter leg.
Perimeter of the 30-60-90 Triangle
The perimeter of a triangle is the sum of the lengths of its sides. In the case of the 30-60-90 triangle, we can use the fixed ratio of the sides to find its perimeter.
Let’s assume that the shortest side of the triangle has a length of “a”. Then, the length of the side opposite the 60-degree angle is “a√3”, and the length of the hypotenuse is “2a”. Therefore, the perimeter of the 30-60-90 triangle is:
Perimeter = a + a√3 + 2a = (1 + √3 + 2)a
We can simplify this expression by using the approximation of √3 ≈ 1.732, which gives us:
Perimeter ≈ (1 + 1.732 + 2)a ≈ 4.732a
Area of the 30-60-90 Triangle
To find the area of a 30-60-90 triangle, we use the formula A = (1/2)bh, where b is the length of the base and h is the length of the height. In this 30 60 90 triangle formula, the base is the side opposite to the 60-degree angle, which we know is x√3/2, and the height is the side opposite to the 30-degree angle, which we know is x/2.
So, the area of the triangle is:
A = (1/2)(x√3/2)(x/2)
= x^2√3/8
We can also express the area in terms of the hypotenuse, using the fact that the hypotenuse is twice the length of the side opposite the 30-degree angle:
A = (1/2)(x/2)(x√3)
= x^2√3/8
Therefore, the area of a 30-60-90 triangle is equal to one-eighth of the square of the hypotenuse multiplied by √3.
30-60-90 Triangle Theorem
The 30-60-90 triangle theorem states that in a right triangle where one of the angles measures 30 degrees, the ratio of the length of the hypotenuse to the length of the shorter leg is 2:1, and the ratio of the length of the longer leg to the length of the shorter leg is √3:1. In other words, if the shorter leg is denoted as x, the longer leg will be x√3, and the hypotenuse will be 2x.
To illustrate this theorem, consider a right triangle ABC with a right angle at vertex C. Let angle B measure 60 degrees, and angle A measure 30 degrees. If side AB has a length of x, then using the trigonometric ratios, we can determine that side AC will have a length of x√3 and side BC will have a length of 2x.
This theorem is based on the ratios of the sides of a triangle and enables us to find the length of unknown sides of a right triangle without the use of trigonometric functions or complicated calculations. We can simply use the ratios provided by the 30-60-90 triangle theorem to solve for the unknown side lengths.
Solved Examples on 30-60-90 Triangle
Here are some 30-60-90 triangle examples:
Example 1:
In a 30-60-90 triangle, the length of the shorter leg is x. What is the length of the hypotenuse?
Solution:
In a 30-60-90 triangle, the ratio of the lengths of the sides is x: x√3: 2x.
Therefore, the length of the hypotenuse is 2x.
Answer: The length of the hypotenuse is 2x.
Example 2:
In a 30-60-90 triangle, the length of the hypotenuse is 10 cm. What is the length of the longer leg?
Solution:
In a 30-60-90 triangle, the ratio of the lengths of the sides is x: x√3: 2x.
Therefore, the length of the longer leg is x√3.
If the hypotenuse is 10 cm, then 2x = 10 cm, so x = 5 cm.
Thus, the length of the longer leg is 5√3 cm.
Answer: The length of the longer leg is 5√3 cm.
Example 3:
In a 30-60-90 triangle, the longer leg is 4 cm. What is the length of the hypotenuse?
Solution:
In a 30-60-90 triangle, the ratio of the sides lengths’ is x: x√3: 2x.
Therefore, if the longer leg is 4 cm, then x√3 = 4√3 cm and 2x = 8 cm.
Thus, the length of the hypotenuse is 8 cm.
Answer: The length of the hypotenuse is 8 cm.
We recommend using Brighterly Worksheets for the topic “30-60-90 triangles”. These worksheets provide you with clear visual aids and practice problems that will help you better understand this topic and improve your problem-solving skills.
Frequently Asked Questions
How is the 30-60-90 triangle similar to 45-45-90?
The 30-60-90 triangle and the 45-45-90 triangle are both special right triangles with specific relationships between their side lengths. Although they have different angles, the ratio of the sides in a 30-60-90 triangle is the same as the ratio of the legs in a 45-45-90 triangle.
Which side is the longer leg of a 30-60-90 triangle?
In a 30-60-90 triangle, the longer leg is opposite the 60-degree angle. The shorter leg is opposite to the 30-degree angle, and the hypotenuse is opposite to the 90-degree angle. So, the longer leg is opposite to the angle that measures 60 degrees.
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