The altitude of an equilateral triangle is 9 meters, what is the side length

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electreto05's profile pic

electreto05 | College Teacher | (Level 1) Assistant Educator

Posted on

Consider that:

1. The triangle has three sides equal, so that each inner angle is 60°

2. The altitude, forms a right angle to the opposite side, so that there are two new right triangles whose interior angles are 90°, 60° and 30°

For each angle of 60°, in the new triangles, we can write that:

sine 60°= opposite side/hypotenuse


opposite side = Altitude of the equilateral triangle.

hypotenuse = side of the equilateral triangle (what we are looking)

Then we can write:

sine 60°= 9/hypotenuse

hypotenuse = side of the equilateral triangle = 9/0.866 = 10.39 m

sciftw's profile pic

sciftw | High School Teacher | (Level 1) Educator Emeritus

Posted on

The formula for that solution is 

A = (2/sqrt of 3) x h 

I couldn't figure out the formula button on this website.  

A = side length.  Since you have an equilateral triangle, all of the sides will be equal. 

h = altitude of your triangle, which you stated is 9 meters

sqrt = square root.  

So with your numbers, it looks like this

A = (2/sqrt of 3) x 9

A = 10.3923 meters.  

Once you know "A" you can figure out a lot of other stuff about the triangle.  Multiply it by 3 to get the perimeter. 

The semiperimeter of the triangle would be (3A)/2

The area would be 1/4 * sqrt 3 * A^2  

Of course those formulas only work as long as your triangles continue to be equilateral triangles.  

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