30 60 90 triangle calculator (2023)

They form onetriangleas a geometric shape with three sides and three corners. Theangle ratiotells us what kind of triangle it is. In addition tothe 45 45 90 triangle calculator,you can use this calculatorCalculate 30 60 90 triangles. If we talk about calculators as tools, they make checking the accuracy of calculations much easier. So if you want to know more about what this new calculator brings and what formulas you can use, take a look below.

Suppose you want to expand your knowledge. In that case, don't miss these calculators either:Distance between two parallel lines,segment addition calculator, AndKofunktionsrechner – sin, cos, tan, cot, sec, csc,phase shift calculator.

30 60 90 right triangle set

We can define three basic properties and characteristics to understand the 30-60-90 triangle set. These properties are also expressly described forright triangles. The first property says that the longest side of a triangle is calledthe hypotenusehas in practice always twice the length of the shorter leg. You can also calculate the value of the length of thelonger legMultiply the value of theshorter legby the root number 3. And if you know the values ​​of any side 30-60-90 of the triangle, it will help you to find the values ​​of the unknown sides.

30 60 90 triangle calculator (1)

You can mark the value of the shorter leg as y, the value of the hypotenuse as 2y, while the value of the longer leg is:

y\cdot \sqrt{3}

We can emphasize that all triangles 30-60-90 are similar and that two such triangles sharing one long leg thus form an equilateral triangle. Interesting, isn't it?

30 60 90 triangle calculator (2)

The figure below shows an example of a 30-60-90 right triangle whose side lengths should always match. Triangle ABC, angle C =30 Grad, Store A =60 Gradand angle B =90 Grad. The side of the triangle is opposite the angle of 30 degrees,AB = y, and as such it is called short leg because it is also the shortest. The side marked opposite the angle of 60 degreesBC = y√3is called the long leg of the triangle, and it is the middle side in length. At the same time, the remaining side of the triangle is also marked the longestAC = 2 years, known as the hypotenuse.

30 60 90 triangle calculator (3)

30 60 90 triangle ratio

When determining the 30-60-90 triangle set, it's important to know itratio of sidesof the triangle 30-60-90. Since these are a special type of right triangle, you must always align the lengths of the sides of the triangle so that the ratio of 30-60-90 triangles looks like this:

for pages

1 : √3 : 2 = (y : y√3 : 2y)

or for angles

1 : 2 : 3 = (30° : 60° : 90°)

30 60 90 triangle formula

From a geometric point of view, the formula used for the calculation lies in the basics of trigonometry. If we want to calculate side lengths, we can use a sineStoreof 30 degrees. According to the attached graphic, we can calculate the value of the side lengths b and c using several formulas:

\frac{a}{c} = sin30 = \frac{1}{2}

Woc = 2a

\frac{b}{c} = sin60 = \frac{\sqrt{3}}{2}

Wob = c√3/2

30 60 90 triangle calculator (4)

30 60 90 triangle rules and properties

When it comes to the rules you can apply in the 30-60-90 triangle, you will likely manipulate the valuesthe hypotenuseand the shorter legs of the triangle. The geometry is behind all these connections and connections between all three sides of the triangle, and you can see what it looks like in the table below:

If you know…Then…To get…
HypotenuseDivide by 2short leg
short legMultiply by 2Hypotenuse
short legMultiply by √3long leg
long legDivide by √3short leg

Following the properties and rules, one can lose sight of the fact that it's still a triangle. It would help if you never forget that the shape of a triangle is constant under all conditions. Therefore, interior angles should be added180 Grad, while the sides should follow the rules ofPythagorean theorem:

a^{2} + b^{2} = c^{2}

How do we solve a 30 60 90 triangle? - Example

Let us give you a more detailed explanation of the formulas used to calculate each of the cases listed below, and then we will apply the same formulas to a case study.

Solve Part A

If you need the formula to calculate the value of the length of leg A, we can rely on the value of leg b according to the following expression:

a = \frac{b\cdot \sqrt{3}}{3}

Solve Part B

After that, you need to calculate the length of leg B using a formula that looks like this:

b = a\sqrt{3}

Calculate the hypotenuse

Follow the previous explanations and rules related to the longest side of the 30-60-90 triangle, the hypotenuse.
The formula for calculating the hypotenuse looks effortless:

c = 2 \cdot a

Area of ​​a 30-60-90 triangle

The basic formula for calculating theAreaof a two-dimensional shape is the number of square units you need to fill the space inside that shape. In the case of 30-60-90 triangles, the formula you can use to calculate the area of ​​a triangle is:

A = \frac{1}{2}\cdot b\cdot h

where the values ​​are:

A= triangular area
B= base of the triangle
X= height of the triangle

Calculate perimeter

When calculating the perimeter of a triangle of any shape, we need the sum of the edges. therefore, theScopeof the triangle 30-60-90 is equal to the sum of the sides and the hypotenuse. It looks like this:

P = a + b + c


Now that you are familiar with all the rules, formulas and definitions, you can apply this to the following task:

Find the value of the length of the hypnotic of a right triangle when the value of one side is 8 and the other is 8√3.

8: 8√3 = 1: 1√3

From this mathematical expression we conclude that it was 30-60-90. So we know that the length of the hypotenuse is equal to the value of the small side multiplied by the number 2. From this we can conclude that the hypotenuse value is equal to 16 in this example.

To make the calculation easier, you can use our calculator, which works according to an elementary principle. After accessing ourcalculator, you will see special empty name fields. Regardless of what value you enter, whether side a, side b, or triangular area, the calculator will display results for other blank fields.


Which triangle is a 30 60 90 triangle?

A triangle containing angles of 30, 60, and 90 degrees is a 30-60-90 triangle. It represents a special form of the right triangle, since the angles here are in the ratio 1:2:3.

How do you recognize a 30 60 90 triangle?

To identify this triangle, you need to know the basic rules such that the most extended side is twice the length of the shorter side opposite the 30 degree angle and that the side opposite the 60 degree angle is equal to the shorter leg times the square root of the number 3.


What are the remaining sides of a 30-60-90 triangle if the shortest side is 8? ›

And because this is a 30-60-90 triangle, and we were told that the shortest side is 8, the hypotenuse must be 16 and the missing side must be 8 * 3 , or 8 3 . Our final answer is 8√3.

What is the formula for the triangle ratio? ›

Triangle ratio formula

α + β + γ = 180° . From this you can determine first the unknown x , and then the angles: ax , bx , cx . In the next section, we translate these considerations into a step-by-step guide on how to find missing angles in triangles using ratios.

What is the rule for a 30 60 triangle? ›

30°-60°-90° Triangles

In a 30°−60°−90° triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg. To see why this is so, note that by the Converse of the Pythagorean Theorem, these values make the triangle a right triangle.

What is the formula for the 90 triangle? ›

The formula is a2+b2=c2 a 2 + b 2 = c 2 where a and b are the legs of the right triangle and c is the hypotenuse.

Which is the longest side in a 30-60-90 triangle *? ›

30-60-90 Triangle Properties. The sides of a 30-60-90 triangle are identified by their relation to the angles. The side opposite the 30-degree angle is called the shorter leg. The side opposite the 60-degree angle is called the longer leg.

How do you find the 3rd side of a triangle? ›

Pythagoras theorem: In a right triangle, if hypotenuse, perpendicular and base are its sides, then as per the theorem, the square of hypotenuse side is equal to the sum of the square of base and square of perpendicular. Hence, if we know any two sides, then we can easily find the third side of the triangle.

What is the 3 4 5 ratio triangles? ›

The 3:4:5 triangle is the best way I know to determine with absolutely certainty that an angle is 90 degrees. This rule says that if one side of a triangle measures 3 and the adjacent side measures 4, then the diagonal between those two points must measure 5 in order for it to be a right triangle.

Why does every 30-60-90 triangle have the same side ratios? ›

A 30-60-90 triangle is a right triangle with angle measures of 30º, 60º, and 90º (the right angle). Because the angles are always in that ratio, the sides are also always in the same ratio to each other.

What is a 30-60-90 triangle math? ›

What Is a “30-60-90” Triangle? 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° angle is always the smallest (shortest leg).

Can you solve a triangle with no side lengths? ›

This means we are given all three angles of a triangle, but no sides. AAA triangles are impossible to solve further since there are is nothing to show us size ... we know the shape but not how big it is.

What triangle is 45 45 90 and 30 60 90? ›

30-60-90 Triangle

In an isosceles right triangle, the angle measures are 45°-45°-90°, and the side lengths create a ratio where the measure of the hypotenuse is sqrt(2) times the measure of each leg as seen in the diagram below.

What is the 45 45 90 rule? ›

The 45-45-90 triangle rule states that the three sides of the triangle are in the ratio 1:1:\(\sqrt{2}\). So, if the measure of the two congruent sides of such a triangle is x each, then the three sides will be x, x and \(\sqrt{2}x\). This rule can be proved by applying the Pythagorean theorem.

Is a Pythagorean triple a 30-60-90 triangle? ›

Because at least one side of a 30°- 60°- 90° triangle must contain a square root, a 30°- 60°- 90° triangle cannot belong to any of the Pythagorean triple triangle families.

How do you find the sides of a triangle with only angles? ›

If A, B, C are angles with A+B+C=180∘, then sinA, sinB, sinC are the lengths of the sides of a (not the!)
Assuming A,B,C form a triangle.
  1. cos(A+B+C)=-1, cos(A+B)cos(C)-sin(A+B)sin(C)=-1.
  2. sin(A+B+C)=0, sin(A+B)cos(C)+cos(A+B)sin(C)=0.
  3. sin(A+B)=sin(A)cos(B)+cos(A)sin(B).
  4. cos(A+B)=cos(A)cos(B)-sin(A)sin(B).
Apr 6, 2014

What is a 30 60 90 plan? ›

A 30-60-90 day plan is a document used to set goals and strategize your first three months in a new job . 30-60-90 day plans help maximize work output in the first 90 days in a new position by creating specific, manageable goals tied to the company's mission and the role's duties and expectations.

Is the longer leg of a 30-60-90 triangle half the hypotenuse always sometimes never? ›

The short leg of a 30-60-90 triangle is always 1/2 the length of the hypotenuse. You could also switch it around and say that the hypotenuse is always twice the length of the short leg. Remember, a 30-60-90 triangle is half of an equilateral triangle.

Does Pythagorean theorem work on all triangles? ›

Hence we can say that the Pythagorean theorem only works for right triangles.

How do you find the lengths of the other two sides of a right triangle? ›

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, a2=b2+c2 a 2 = b 2 + c 2 , where a is the hypotenuse and b,c are the other two sides. This is also called the Pythagorean Theorem.

How do you find the missing 3 side of a triangle? ›

🙋 Our Pythagorean theorem calculator will help you if you have any doubts at this point. Apply the law of sines or trigonometry to find the right triangle side lengths: a = c × sin(α) or a = c × cos(β) b = c × sin(β) or b = c × cos(α)

Can you find the missing number triangle answer? ›

The answer to this "Missing Number in Triangle Puzzle", can be viewed by clicking on the button. Please do give your best try before looking at the answer. The Answer is 4. Multiplying downside corner numbers of the triangle and then subtract the upper corner number to get the central number.


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