Perimeter of an Equilateral Triangle: Formula and Examples

Perimeter of an equilateral triangle is the total length around the triangle.

An equilateral triangle is a triangle in which all three sides are equal. That is why finding its perimeter is quite simple: you just need to add the lengths of all three sides.

In this article, we will look at what the perimeter of an equilateral triangle is, write down the main formula, and use it to solve practical problems.

Perimeter of an Equilateral Triangle: Main Formula

To calculate the perimeter of an equilateral triangle, you need to add the lengths of its three sides.

Since all sides of an equilateral triangle have the same length, it is enough to multiply the length of one side by \( 3 \). Therefore, for triangle \( ABC \) shown above, we have:

\[
P=3\cdot AB,
\]

where \( P \) is the perimeter of the equilateral triangle.

Note. If we denote the side length of an equilateral triangle by the letter \( a \), then the perimeter formula takes the more familiar form:
\[
P=3\cdot a.
\]

Perimeter of an Equilateral Triangle: Examples with Answers

The examples below are solved using the perimeter formula for an equilateral triangle. Try to do the calculations on your own first, and then compare your results with the answers provided.

Example 1. Find the perimeter of an equilateral triangle if its side is \( 4 \) cm

We are given that the side of the triangle is \( 4 \) cm. Then, using the perimeter formula, we get:

\[
P=3\cdot a=3\cdot 4=12.
\]

So, the perimeter of the equilateral triangle is \( 12 \) cm.

Example 2. Find the perimeter of an equilateral triangle if each side is \( 8 \) cm

In this case, each side of the triangle has a length of \( 8 \) cm. Let us substitute this value into the perimeter formula:

\[
P=3\cdot a=3\cdot 8=24.
\]

So, the perimeter of the equilateral triangle is \( 24 \) cm.

Example 3. The perimeter of an equilateral triangle is \( 27 \) cm. Find the side of the triangle

In this example, the perimeter of the equilateral triangle is known, and we need to find the length of its side. Let us use the same formula:

\[
P=3\cdot a,\qquad 27=3\cdot a,\qquad a=9.
\]

So, the side length of the equilateral triangle is \( 9 \) cm.

Example 4. Find the perimeter of an equilateral triangle if its area is \( 56\ \text{cm}^2 \)

We know that the area of an equilateral triangle is calculated using the formula

\[
S=\frac{\sqrt{3}\cdot a^2}{4}.
\]

So, first let us find the side of the triangle by substituting the known area value into the formula:

\[
56=\frac{\sqrt{3}\cdot a^2}{4},\qquad 224=\sqrt{3}\cdot a^2,\qquad a\approx 11.372.
\]

Now that the side length is known, let us substitute it into the perimeter formula:

\[
P=3\cdot a=3\cdot 11.372\approx 34.116.
\]

Thus, the perimeter of the equilateral triangle is \( 34.116 \) cm.

Example 5. Find the perimeter of an equilateral triangle if its height is \( 9 \) cm

To find the perimeter, we first need to determine the side length of the triangle. For this, let us use the formula for the height of an equilateral triangle:

\[
h=\frac{\sqrt{3}\cdot a}{2},\qquad 9=\frac{\sqrt{3}\cdot a}{2},\qquad a\approx 10.392.
\]

Next, let us substitute the side length we found into the perimeter formula:

\[
P=3\cdot a=3\cdot 10.392\approx 31.176.
\]

So, the perimeter of the equilateral triangle is \( 31.176 \) cm.

What to Read Next: Helpful Topics to Explore

After learning the perimeter formula, it is worth exploring other topics related to the equilateral triangle as well. They will help you better understand the properties of this figure and see how different formulas are connected.

1. Equilateral Triangle: Definition and Properties — Learn what properties an equilateral triangle has and what makes it special among other triangles.
2. Height of an Equilateral Triangle: Formulas and Examples — See how the height of an equilateral triangle is calculated and how this formula is applied in practice.
3. Area of an Equilateral Triangle: Formulas and Examples — Learn how to find the area of an equilateral triangle and how to use this formula correctly in problems.

Perimeter in Code: Try Writing Your Own Program

This topic can also be explored through programming. If you are interested not only in using ready-made formulas but also in turning them into code, try writing a small program in your favorite programming language that calculates the perimeter of an equilateral triangle from the length of its side. This approach helps you see the practical value of geometry more clearly, because the formula stops being just a rule from a textbook and becomes a clear algorithm that you can implement on your own. It is also a great opportunity to combine mathematics and programming and test your knowledge with a simple and clear example.