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Area of an Equilateral Triangle — Formula & Examples

Area of an Equilateral Triangle

The formula is given below.

 

Equilateral Triangle

s = length of a side

Formula: Area equals s squared times square root of 3, divided by 4

 Equilateral triangle with all three sides labeled "s", indicating equal side lengths.

 

See also

Equilateral triangle

Key Formula

A=34 s2A = \frac{\sqrt{3}}{4}\,s^2
Where:
  • AA = Area of the equilateral triangle
  • ss = Length of one side of the equilateral triangle

Worked Example

Problem: Find the area of an equilateral triangle with a side length of 6 cm.
Step 1: Write the formula for the area of an equilateral triangle.
A=34 s2A = \frac{\sqrt{3}}{4}\,s^2
Step 2: Substitute s = 6 into the formula.
A=34 (6)2=34⋅36A = \frac{\sqrt{3}}{4}\,(6)^2 = \frac{\sqrt{3}}{4} \cdot 36
Step 3: Simplify by dividing 36 by 4.
A=93A = 9\sqrt{3}
Step 4: Approximate the result using √3 ≈ 1.732.
A≈9×1.732=15.588 cm2A \approx 9 \times 1.732 = 15.588 \text{ cm}^2
Answer: The area is 9√3 ≈ 15.59 cm².

Another Example

Problem: An equilateral triangle has a side length of 10 m. What is its area?
Step 1: Apply the formula with s = 10.
A=34 (10)2=34⋅100A = \frac{\sqrt{3}}{4}\,(10)^2 = \frac{\sqrt{3}}{4} \cdot 100
Step 2: Simplify.
A=253A = 25\sqrt{3}
Step 3: Approximate the value.
A≈25×1.732=43.30 m2A \approx 25 \times 1.732 = 43.30 \text{ m}^2
Answer: The area is 25√3 ≈ 43.30 m².

Frequently Asked Questions

Where does the formula (√3/4)s² come from?
Start with the general triangle area formula: A = ½ × base × height. For an equilateral triangle with side s, the base is s. The height splits the triangle into two 30-60-90 right triangles, giving a height of (√3/2)s. Substituting: A = ½ × s × (√3/2)s = (√3/4)s².
How do I find the area of an equilateral triangle if I only know the height?
Since the height h = (√3/2)s, you can solve for the side: s = 2h/√3. Then substitute back into the area formula. Alternatively, use A = h²/√3 directly, which comes from combining both formulas.

Area of an equilateral triangle vs. Area of a general triangle

The general triangle area formula is A = ½ × base × height, which requires you to know (or compute) the height. For an equilateral triangle, all sides are equal, so the height is always (√3/2)s. This lets you collapse the formula into A = (√3/4)s², needing only the side length. The equilateral formula is a special case of the general one — it's faster when you know the triangle is equilateral.

Why It Matters

Equilateral triangles appear frequently in engineering, architecture, and tiling patterns because of their perfect symmetry. Knowing this formula lets you quickly compute areas in problems involving regular hexagons (which are made of six equilateral triangles), truss structures, and tessellations. It also reinforces how special properties of a shape — here, equal sides — can simplify a general formula into something much more convenient.

Common Mistakes

Mistake: Using ½ × s × s instead of (√3/4)s², treating the side as both the base and the height.
Correction: The height of an equilateral triangle is not equal to its side. The height is (√3/2)s, which is shorter than s. Always use the correct height or apply the dedicated formula.
Mistake: Forgetting to square the side length, writing A = (√3/4)s instead of (√3/4)s².
Correction: Area is measured in square units, so the side length must be squared. Double-check that s is raised to the second power.

Related Terms