How To Find The Height Of An Equilateral Triangle Using The Pythagorean Theorem?

Finding the height of an equilateral triangle using the Pythagorean theorem involves a few steps that leverage the unique properties of this special type of triangle.

Understanding the Properties

An equilateral triangle has three equal sides and angles of 60 degrees. The altitude, or height (hh

ℎ

), is a line from a vertex perpendicular to the opposite side. In an equilateral triangle, this altitude also acts as the median and angle bisector.

Steps to find the height

1. Draw the altitude: Drawing the altitude from one vertex to the opposite side's midpoint creates two congruent right-angled triangles.

2. Identify the sides: In one of these right-angled triangles, the hypotenuse is the equilateral triangle's side length (aa

   𝑎

   ). One leg is half the base of the equilateral triangle (a/2a / 2

   𝑎/2

   ), and the other leg is the height (hh

   ℎ

   ).

3. Apply the Pythagorean theorem: The theorem states that in a right triangle, a2+b2=c2a squared plus b squared equals c squared

   𝑎2+𝑏2=𝑐2

   , where cc

   𝑐

   is the hypotenuse. Substituting the side lengths of our right triangle (a/2a / 2

   𝑎/2

   , hh

   ℎ

   , and aa

   𝑎

   ), we get (a/2)2+h2=a2open paren a / 2 close paren squared plus h squared equals a squared

   (𝑎/2)2+ℎ2=𝑎2

   .

4. **Solve for hh

   ℎ

   :** Solving the equation for hh

   ℎ

   :

   • a2/4+h2=a2a squared / 4 plus h squared equals a squared

     𝑎2/4+ℎ2=𝑎2

   • h2=a2−a2/4=3a2/4h squared equals a squared minus a squared / 4 equals 3 a squared / 4

     ℎ2=𝑎2−𝑎2/4=3𝑎2/4

   • Taking the square root gives h=3a2/4=(a3)/2h equals the square root of 3 a squared / 4 end-root equals open paren a the square root of 3 end-root close paren / 2

     ℎ=3𝑎2/4√=(𝑎3√)/2

     .

Thus, the height of an equilateral triangle with side length aa

𝑎

is given by the formula h=(a3)/2h equals open paren a the square root of 3 end-root close paren / 2

ℎ=(𝑎3√)/2

.

Example

For an equilateral triangle with a side length of 10 units:

• Using the formula: h=(103)/2h equals open paren 10 the square root of 3 end-root close paren / 2

  ℎ=(103√)/2

  .

• Simplifying: h=53h equals 5 the square root of 3 end-root

  ℎ=53√

  .

• The approximate height is h≈8.66h is approximately equal to 8.66

  ℎ≈8.66

  units.

This process demonstrates how the Pythagorean theorem is used to derive a specific formula for the height of an equilateral triangle.