What Is the Altitude of an Equilateral Triangle?

The altitude of an equilateral triangle is the perpendicular line from a vertex to the opposite side.

You probably know the formula h = (a√3)/2 by heart. But that √3 isn’t pulled from thin air — it comes from the 30-60-90 right triangle hidden inside every equilateral shape. Understanding where the formula comes from makes it easier to use and even easier to remember.

This article walks through the definition, the derivation, and two ways to apply the height formula. You’ll see step-by-step examples, a reverse calculation, and a quick reference table you can use for homework or test prep.

What Exactly Is an Altitude?

An altitude is a line segment drawn from a vertex straight down to the opposite side, forming a perfect 90-degree angle with that side. In an equilateral triangle — where all three sides are the same length — there are three altitudes, one from each vertex.

All three altitudes are equal in length. That makes sense because the triangle is perfectly symmetric. Draw any altitude, and you split the equilateral triangle into two mirror-image right triangles.

Those right triangles are special: they’re 30-60-90 triangles. The altitude itself becomes the longer leg, the base gets cut in half, and the original side length becomes the hypotenuse. That relationship is the key to the formula.

Why the Altitude, Median, and Angle Bisector Are the Same

Most triangles keep these lines separate. In an equilateral triangle, they all collapse into one — a quirk that simplifies geometry problems. Here’s what that means for each line:

• Altitude: A perpendicular line from vertex to base. Meets the base at a right angle.
• Median: A line from vertex to the midpoint of the opposite side. In an equilateral triangle, the altitude and median are the same segment.
• Angle bisector: A line that splits the vertex angle into two equal 30-degree halves. It also lands on the same line.
• Perpendicular bisector: A line that cuts the base in half at a right angle. Because the triangle is symmetric, this is exactly the same as the altitude from the top vertex.

That means you only ever need to draw one line from a vertex. It does all four jobs at once. This property is the reason the altitude formula is so clean — the base is simply cut in half, and the Pythagorean theorem does the rest.

Deriving the Formula Using the 30-60-90 Triangle

Take an equilateral triangle with side length a. Draw an altitude from the top vertex to the base. That altitude splits the base into two equal parts, each a/2 long. Now you have a right triangle with hypotenuse a, one leg a/2, and the other leg h (the altitude).

Apply the Pythagorean theorem: (a/2)² + h² = a². Simplify to a²/4 + h² = a², so h² = 3a²/4. Take the square root: h = (a√3)/2. That’s the formula.

The derivation shows exactly why the √3 appears. For a more detailed walkthrough of this property, see the altitude of an equilateral triangle explanation on Byjus.

The pattern is clear: for every 2-unit increase in side length, the altitude grows by √3. That linear relationship makes the formula easy to scale once you know one pair.

How to Use the Formula Step by Step

Applying the altitude formula takes only a few steps. Here’s the process you’d follow for any equilateral triangle:

1. Identify the side length. Let’s call it a. Make sure all three sides are equal — if one side is different, the formula doesn’t apply.
2. Plug a into h = (a√3)/2. Write the expression exactly as shown. For example, if a = 8, write h = (8√3)/2.
3. Simplify the fraction. Divide the number by 2. In the example, 8/2 = 4, so h = 4√3.
4. Multiply by √3 for a decimal. Use √3 ≈ 1.732. For a side of 8, h ≈ 4 × 1.732 = 6.928 units.

Try another: if a = 5, then h = (5√3)/2 ≈ (5 × 1.732)/2 = 4.33 units. You can check any of the values in the table above to confirm your work.

Working Backward: Finding Side Length from Altitude

Sometimes you know the altitude and need the side length. The reverse formula comes from solving h = (a√3)/2 for a. Multiply both sides by 2 to get 2h = a√3, then divide by √3: a = (2h)/√3.

For example, if the altitude is 18 inches, the side length is (2 × 18)/√3 = 36/√3 ≈ 20.784 inches. That’s useful when you’re given the height in a problem and have to find the perimeter or area. The height formula equilateral triangle page on Cuemath includes several worked examples of this reverse calculation.

Notice the pattern again: when the altitude triples, the side length also triples. The relationship between height and side length is proportional — no squaring or rooting hides inside this reverse step.

The Bottom Line

The altitude of an equilateral triangle is a perpendicular line from any vertex to the opposite side. Its length follows h = (a√3)/2, a formula that comes directly from the 30-60-90 right triangle formed when you drop that altitude. The same line also serves as the median, angle bisector, and perpendicular bisector — a symmetry unique to equilateral triangles.

If you’re preparing for a geometry exam and want more practice, your school’s math tutor or a Khan Academy geometry module can guide you through problems that use altitude to find area, perimeter, or missing side lengths. The height formula equilateral triangle derivation is a standard curriculum topic worth mastering step by step.