How to Find the Height of an Equilateral Triangle When the Perimeter Is Given?

To find the height of an equilateral triangle when its perimeter is given, you first calculate the side length of the triangle from the perimeter, and then use that side length to determine the height with a specific formula.

An equilateral triangle is a polygon with three equal sides and three equal angles (each 60 degrees). Its unique properties allow for direct calculation of its height once a side length is known.

Step-by-Step Guide to Calculating the Height

Finding the height involves two straightforward steps:

1. Calculate the Side Length (a) from the Perimeter (P):
   The perimeter of an equilateral triangle is the sum of its three equal sides. If 'a' represents the length of one side, then the perimeter P = a + a + a, which simplifies to P = 3a.
   Therefore, to find the side length, you divide the given perimeter by 3:

   • a = P / 3

2. Calculate the Height (h) using the Side Length (a):
   Once the side length 'a' is known, you can find the height using the formula derived from the Pythagorean theorem or properties of special right triangles (30-60-90 triangle). The height divides an equilateral triangle into two congruent 30-60-90 right-angled triangles.
   The formula for the height 'h' of an equilateral triangle is:

   • h = (√3 / 2) * a
     This can also be written as h = ½(√3a).

Essential Formulas for Equilateral Triangles

Here's a quick reference for the formulas involved:

Aspect	Formula	Description
Perimeter (P)	P = 3a	Sum of the lengths of all three equal sides.
Side Length (a)	a = P / 3	Derived from the perimeter, useful when P is known.
Height (h)	h = (√3 / 2) * a or h = ½(√3a)	The perpendicular distance from a vertex to the opposite side.
Area (A)	A = (√3 / 4) * a²	The space enclosed by the triangle.

Practical Example

Let's work through an example to illustrate the process:

Question: An equilateral triangle has a perimeter of 30 cm. Find its height.

Solution:

1. Find the side length (a):

   • Given Perimeter (P) = 30 cm
   • a = P / 3
   • a = 30 cm / 3
   • a = 10 cm

2. Find the height (h):

   • Using the side length a = 10 cm
   • h = (√3 / 2) * a
   • h = (√3 / 2) * 10 cm
   • h = 5√3 cm

   To get a numerical approximation:

   • h ≈ 5 * 1.732
   • h ≈ 8.66 cm

Therefore, the height of the equilateral triangle with a perimeter of 30 cm is 5√3 cm (approximately 8.66 cm).

This method provides a reliable and precise way to determine the height of any equilateral triangle when only its perimeter is given.