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​Video reviews the rules of 30-60-90 triangles.
Also Solves: Find the hypotenuse and long side of a triangle from a short side of 3 units.
Find the short side of a 30-60-90 triangle given a hypotenuse of 10 units.
Find the hypotenuse given a long side of 6 units.

30-60-90 Triangles are classified as "special right triangles". They are special because of special relationships among the triangle legs that allow one to easily arrive at the length of the sides with exact answers instead of decimal approximations when using trig functions.

30 60 90 triangle examples with answers

30 60 90 triangle rules

30 60 90 triangle characteristics

30 60 90 triangle completes an arithmetic progression
30+30=60+30 =90

An arithmetic progression is a sequence of numbers in which the difference of any two successive numbers is a constant.

For instance, 2,4,6,8 is an arithmetic progression with a constant of 2

30 60 90 triangle is formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles.

The following diagram summarizes the rules of a 30-60 90 Triangle
Common Core Standard G.SRT.6
Keywords:Side ratios,properties of angles,right triangles,trigonometric ratios
​Questions answered in this video

If given the hypotenuse of a 30 60 90 triangle,how do you find the leg length?

If given the leg length of a thirty sixty ninety triangle, how do you find the hypotenuse?

What shortcuts can I use to find the leg lengths and hypotenuse of a special right triangle easily?

Shortcuts for side lengths of a 30-60-90

equilateral triangle
finding short side 30 60 90
30 60 90 triangle

30 60 90 Triangles

E1. Find the hypotenuse of a 30-60-90 triangle with a short side of 3 units

Hypotenuse=  Step 1.   Use the formula   2*s 
                          Step 2.  2*3 =6 units
E2. Find the long side of a thirty sixty ninty triangle with a short side of 3 units.

Long leg = Step 1. Use the formula  short side√(3 ) 
                    Step 2. 3√3 units
E3. Find the short side of a right triangle with a hypotenuse of 10 units.

Short side = Step 1.  Use the formula 1/2 * hypotenuse
                       Step 2. 1/2*10=5 units
E4. Find the short side of a thirty sixty ninty triangle with a long leg of 6 units.

Short Leg
is always opposite the 30 degree angle.
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special right triangles
45-45-90 triangle
Hypotenuse is always opposite the right angle.
x = the short side
 Isolate x by dividing by square root 3.
Simplify to find the length of the short side.
Rationalize the denominator by dividing by square root 3.
30 60 90 triangle infographic
Equilateral triangle
We all enjoy shortcuts. Use these shortcuts to solve for the hypotenuse, short leg, and long leg lengths.

  • If you know the hypotenuse length divide by two for the short leg length.
  • If you know the short leg length multiply by two for the hypotenuse length.
  • If you know the short leg then multiply by √3 for the long leg length
  • If you know the long leg length divide by √3 for the short leg length
The area of a 30-60-90 triangle equals 1/2base * height. Use the short leg as the base. and the long leg as the height.
A thirty, sixty, ninety, triangle creates the following ratio between the angles and side length.
  • The side opposite the 30 degree angle equals x 
  • The side opposite the 60 degree angle is square root three 
  • The side opposite the 90 degree equals 2x

​Angle-Side Ratios
30   :   60  :  90

x   :    x√3   :  2x

Why is this ratio important ? It allows you to quickly find the side length of a 30:60:90 triangle.

For example, find the length of the hypotenuse of a 30-60-90 triangle with a short side of 4 units. 

Solution, the hypotenuse is always opposite the 90 degree angle.
Just multiply the length of the short side ( x) by 2
4*2 = 8 units.

Given a hypotenuse of 6 units find the long and short legs.

Use the angle-side ratios

Short leg = 6/2 = 3 units

Long leg = 3√3 = 5.1962 units
Long Leg is always opposite the 60 degree angle.