The Ultimate Guide to 30 60 90 Triangle

The 30 60 90 Triangle is a special type of triangle that explore the properties of triangles, right angles and the triangle inequality theorem for answers.

The triangle is one of the most studied shapes in mathematics. It plays a major role in High school, colleges and even SAT exams. There are different types of triangles that students will have to deal with. These triangles include the Equilateral triangle, Isosceles triangle, Acute Triangle and Obtuse Triangle. All of these angles have their uniqueness. However, computing them is generally restrictive to borderline triangle calculations.

A special triangle that can explore much more than the above triangle measured is the 30 60 90 triangles. Geometrically, the 30 60 90 triangle is basically a right-angle triangle

However, it is not just any type of right angle. The right angle must have three angels that are shared as 30⁰, 60⁰ and 90. This article will provide a comprehensive understanding of the 30 60, and 90 triangles so that students can understand it and easily score high whenever they have to handle them during SAT.

We will consider the impact of the Triangle Inequality Theorem and the Right angle property to arrive at our answer easily.

What is a 30-60-90 Triangle?

The 30 60 and 90 Triangle is a special right angle with three angles given as 30⁰, 60⁰ and 90⁰. This angles are constant and will always have his three angles consistently as 30⁰, 60⁰ and 90⁰. There is no dedicated 30 60 90 triangle formula as it is more of logic and theory.

Now let us consider our right angle in the first section.

How to Know the Longest side of a Triangle: Triangle Inequality Theorem

In Mathematics, it is important to understand that the side opposite the largest angle of a triangle is always the largest side. Even in our study of the triangle inequality theorem, we could consistently show that the largest side always lies opposite against the longest angle.

Geometrically, it is easy to show this. We will consider a triangle below

Based on the philosophy of the right angle, the bottom angle of the side a will always be 90⁰. It is the only angle represented with a square instead of a curve. Now, side b, which is the hypotenuse, is always assumed to be the longest side of any right angle. This is because the other two angles can never get up to 90⁰

Prior to now, students may not have known why the side b was always the longest side. It is because it is the side directly opposite side a where the biggest 90⁰ angle is based. The second longest side will be side a if the angle between the intersection of b and c (it is the angle opposite side a) is the second largest. Side b in the same vein will instead take the position of side a if the angle between the intersection of a and b (it is the angle opposite side c) is the second largest.

The above is the way to know which side is the longest side of a triangle.

There are some points we need to note to get the right answer

No 1: We already know the square angle is 90⁰ above. Now the 60⁰ and 30⁰ will have to be any of the two angles. For the 30⁰ 60⁰ and 90⁰ triangle, the basic position of the angles is always

N0 2: From the above triangle, we now know where each angles should be placed in our triangle.

Properties of the 30 60 and 90 Triangle

Now there are certain things that we need to know about the above the triangle.

Property 1

Side c is the shortest angle of the triangle and this is because it is the side opposite the shortest angle of 30⁰. Side a follows as it is the side opposite the 60⁰. Side b is the largest side as it is the side opposite the largest angle 90⁰.

As is known about the right angle, the side b is the hypotenuse, side a is the opposite while side c is the adjacent.

Property 2

The length of the adjacent which is side c is given as x

The length of the hypotenuse which is side b is given as 2x

That is, the 30 60 90 triangle is actually

Knowing the rules of the right angle triangle is very important because it helps student boycott too many calculations and gets their answers quite easily whenever they have to deal with this kind of triangle.

Property 3

For every triangle, the entire angle is also 180⁰. Whether it is an equilateral triangle or even the right angle triangle, some of the entire angle is 180⁰. This is why since a right-angle triangle already has 90⁰ at the intersection of the opposite and the adjacent, there is only 90⁰ available for the two other angles to share. In the 30 60 90 triangle, the remaining 90⁰ is shared between 60⁰ and 30⁰.

This property actually helps people find the angles of a right angle as long as one of the non-90⁰ angle is available.

Property 4

It is a must that the adjacent will always be the shortest side. While It depends on the size of the triangles to tell us which side is the shortest among all three, students must make a conscious effort to set 30⁰ at the intersection between the opposite the hypotenuse, which is the same as directly opposite the adjacent. If we have the 30⁰ 60⁰ and 90⁰ in the form below;

Then the shortest side will automatically be the opposite side and the second shortest side will be the adjacent. This type of triangle cannot be considered as a 30 60 90 triangle.

Property 5

The 30 60 and 90 Triangle uses the Pythagoras theorem just like any type of right angle. The Pythagoras theorem formula states that the square of a hypotenuse equals the sum of the squares of its adjacent and opposite.

It is actually given as;

hyp² = adj² + opp²

When is the Right Time to Use the 30 60 90 Triangle Philosophy?

Geometry Problems

In SAT, it is very common to deal with the 30 60 90 triangle. While it may not be totally spelt out that it is the 30 60 90 triangle, based on the properties discussed, it is quite easy to tell when students are presented with such angles.

1. If there is a right angle calculation in which there is one angle available, and it is actually 30⁰ or 60⁰, then such computation is definitely a 30 60 90 Triangle problem. This is because the moment any of the two angles 30⁰ and 60⁰ are available, it is only logical that the other one is the missing piece as the right angle of 90⁰ is always constant.
2.  If there are one angle and two sides available or there are two sides and one angle, we can also solve the right angle triangle with the Pythagoras theorem. 

Trigonometry Problems

When talking about trigonometry problems, we are basically talking about the Sine Cosine and Tangent of trigonometry. There is a relationship between the 30 60 90 Triangle and the trigonometry function.

Now according to trigonometry,

• Sine when expressed in terms of angle, will be

Sin (ϴ of the intersection of opposite and hypotenuse). This is why it is termed as SOH where S is the sine, O means Opposite, and H Means Hypotenuse

• For cosine; it is

Cos (ϴ of the intersection of adjacent and hypotenuse). This is why it is termed as CAH where C is the cosine, A means Adjacent and H Means Hypotenuse

• For Tangent; it is

Tan (ϴ of the intersection of opposite and adjacent). This is why it is termed as TOA where T is the tan, O means Opposite, and A Means Adjacent

Now, if we are given just the angles of a 30 60 90 triangle, we can find the sides with respect to the three SOH CAH TOA of Trigonometry. The answer for each of the trigonometric angles will be the length of the side opposite them. That is how to find the length of the sides using trigonometry

We will consider different examples of the 30 60 90 Triangle

How to Calculate the 30 60 90 Triangle?

The next set of examples will consider how to love the commonly asked SAT questions concerning this special triangle.

Example

Consider the triangle below and find its hypotenuse

Solution

The triangle is clearly a 30 60 90 triangle since two angles are made available i.e 30 and 90. Naturally, 60 is the missing angle.

Now the aim is to find the hypotenuse length.

Clearly, only the opposite has a known length which is given as 56 cm

So how do we find the hypotenuse?

Now recall that for this type of angle;

Adj length = x

Hypotenuse length = 2x

With the above,

Since we know x now, it means we automatically know our adjacent.

By implication,

Hypotenuse length = 2x = 2(34.064) = 68.128 cm

This will make our triangle take the form.

Example

Find the perimeter of the triangle below.

Solution

There is no need to panic seeing the hypotenuse in roots. Students need to follow the same process that has been extensively explained throughout this article to get their answers.

We now have all of the sides of our triangle completed. The next step is to compute the perimeter.

Perimeter of triangle = the sum of all sides

We can calculate the area of the triangle the same we have calculated the perimeter once we have found the sides using the established formulas built on the logic of the 30 60 90 right angle triangle. The bone of contention is simply to find the sides, and the rest is just to use simple established Triangle formulas.

Example

This next example will consider a very common question that comes up on SAT to help students get better information on what they may encounter whenever they have to deal with the 30 60 90 Triangle.

Consider the triangle ABC. What is the length of AD?

Solution

A good look at the above triangle shows that we have two 30 60 90 triangles.

From the left side, the first 30 60 90 triangles are given as BAD, while the second one is on the right side, which is given ad BCD.

Both 30 60 90 triangles are equal to each other, and as such, whatever the length of one will automatically translate to the length of the other.

As such

Since we are looking for AD, which is actually the adjacent of the first 30 60 90 triangle given as ABD, we can simply find the length dc of the second triangle to know it. Length DC is the adjacent of the second 30 50 90 triangle.

We have to use the adjacent of the second triangle because its hypotenuse is readily available.

Clearly 15 cm = 2x

This implies

X = 15/2

As such DC = 15/2

Since DC = 15/2 and DC = AD

Then AD, which is the main length we want to find, is equal to 15/2 = 7.5 cm

Conclusion

The 30 60 90 triangle is actually quite easy to compute when any challenge is brought around it. All that students need to do is just to understand its philosophies and properties. This piece comprehensively outlined everything about the 30 60 90 triangle and all that it stands, and as can be seen, it is quite easy to understand.

All the examples considered in this article are inspired by likely SAT questions and ae among the most difficult 30 60 90 triangles challenges that students will ever face. Each one was comprehensively explained so that students can understand and solve similar related questions.