Equilateral Triangle — Definition, Properties & Examples

Equilateral Triangle

Note: An equilateral triangle is also equiangular. In Euclidean geometry, the angles of an equilateral triangle each measure 60°.

Equilateral Triangle

s = length of a side

Key Formula

A = \frac{\sqrt{3}}{4}\,s^2

Where:

$A$ = Area of the equilateral triangle
$s$ = Length of one side of the triangle

Worked Example

Problem: An equilateral triangle has a side length of 10 cm. Find its area, perimeter, and height.

Step 1: Find the perimeter. Since all three sides are equal, multiply the side length by 3.

P = 3s = 3(10) = 30 \text{ cm}

Step 2: Find the height. Drop a perpendicular from one vertex to the opposite side. This bisects the base into two segments of length s/2. Use the Pythagorean theorem on the resulting right triangle.

h = \sqrt{s^2 - \left(\frac{s}{2}\right)^2} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3} \approx 8.66 \text{ cm}

Step 3: Find the area using the standard formula for an equilateral triangle.

A = \frac{\sqrt{3}}{4}\,s^2 = \frac{\sqrt{3}}{4}(10)^2 = \frac{100\sqrt{3}}{4} = 25\sqrt{3} \approx 43.30 \text{ cm}^2

Step 4: Verify the area using the base-height formula: A = (1/2)(base)(height).

A = \frac{1}{2}(10)(5\sqrt{3}) = 25\sqrt{3} \approx 43.30 \text{ cm}^2 \; \checkmark

Answer: The perimeter is 30 cm, the height is 5√3 ≈ 8.66 cm, and the area is 25√3 ≈ 43.30 cm².

Another Example

This example works backwards from a known area to find the side length, showing how to rearrange the equilateral triangle area formula.

Problem: The area of an equilateral triangle is 48√3 cm². Find the side length and the perimeter.

Step 1: Start with the area formula and substitute the known area.

48\sqrt{3} = \frac{\sqrt{3}}{4}\,s^2

Step 2: Divide both sides by √3 to simplify.

48 = \frac{s^2}{4}

Step 3: Multiply both sides by 4 and take the square root.

s^2 = 192 \quad \Rightarrow \quad s = \sqrt{192} = 8\sqrt{3} \approx 13.86 \text{ cm}

Step 4: Find the perimeter.

P = 3s = 3(8\sqrt{3}) = 24\sqrt{3} \approx 41.57 \text{ cm}

Answer: The side length is 8√3 ≈ 13.86 cm, and the perimeter is 24√3 ≈ 41.57 cm.

Frequently Asked Questions

What is the difference between an equilateral triangle and an isosceles triangle?

An equilateral triangle has all three sides equal, while an isosceles triangle has exactly two sides equal (and the third side different). Every equilateral triangle is technically also isosceles, but not every isosceles triangle is equilateral. The angles in an equilateral triangle are all 60°, whereas an isosceles triangle has two equal angles that are not necessarily 60°.

Why are all angles in an equilateral triangle 60 degrees?

The three interior angles of any triangle sum to 180°. In an equilateral triangle, all three sides are equal, so by the isosceles triangle theorem applied to each pair of sides, all three angles must be equal. Dividing 180° by 3 gives exactly 60° per angle.

How do you find the height of an equilateral triangle?

Drop a perpendicular from any vertex to the opposite side. This creates a right triangle with hypotenuse s and one leg s/2. By the Pythagorean theorem, the height is h = (s√3)/2. For example, a triangle with side length 6 has height (6√3)/2 = 3√3 ≈ 5.20.

Equilateral Triangle vs. Isosceles Triangle

	Equilateral Triangle	Isosceles Triangle
Equal sides	All 3 sides equal	Exactly 2 sides equal
Equal angles	All 3 angles equal (each 60°)	2 angles equal (values vary)
Lines of symmetry	3 lines of symmetry	1 line of symmetry
Area formula	A = (√3/4)s²	A = (1/2) × base × height (no single-variable shortcut)
Special property	Always equiangular	Not equiangular unless also equilateral

Why It Matters

Equilateral triangles appear throughout geometry—from tiling patterns and tessellations to the structure of regular polygons (a regular hexagon is composed of six equilateral triangles). You will use the area formula frequently in standardized tests like the SAT and ACT, as well as in problems involving composite shapes. In engineering and architecture, the equilateral triangle's symmetry gives it uniform strength, making it a fundamental building block of truss structures.

Common Mistakes

Mistake: Using (1/2)(s)(s) as the area instead of (√3/4)s².

Correction: The height of an equilateral triangle is not equal to its side length. The correct height is h = (s√3)/2, which gives the area formula A = (√3/4)s². Using A = (1/2)s² would overestimate the area.

Mistake: Assuming an equilateral triangle can have a right angle.

Correction: Each angle in an equilateral triangle is exactly 60°. No angle can be 90° because the three equal angles must sum to 180°, forcing each to be 60°.

Related Terms

Triangle — General category that includes equilateral triangles
Equiangular Triangle — Equivalent to equilateral in Euclidean geometry
Isosceles Triangle — Has two equal sides; equilateral is a special case
Scalene Triangle — No equal sides; contrast to equilateral
Area of an Equilateral Triangle — Dedicated formula A = (√3/4)s²
Congruent — All three sides are congruent in this triangle
Angle — Each interior angle measures 60°
Side of a Polygon — Equilateral defined by three equal sides