Equilateral Triangle: Definition, Properties, and Formulas
Book A Free Math Class
Equilateral Triangle: Definition, Properties, and Formulas
TL;DR
An equilateral triangle is a triangle with all three sides equal and all three angles equal to 60°. This article covers its definition, properties, the area ( \left( \frac{\sqrt{3}}{4}a^2 \right) ), perimeter ( (3a) ) and height ( \left( \frac{\sqrt{3}}{2}a \right) ) formulas with derivations, six worked examples, and common mistakes.
BT Last updated on July 13, 2026 8 min read
An equilateral triangle is a triangle in which all three sides have the same length. The name says it: equi means "equal" and lateral means "sided." Because equal sides force equal angles, all three interior angles are equal too, and since a triangle's angles add to 180°, each one is exactly ( \frac{180°}{3} = 60° ). So an equilateral triangle is also equiangular.
The equal-side requirement makes it the simplest regular polygon — a regular polygon is one with all sides equal and all angles equal, and the equilateral triangle is the three-sided case. By the end you will know its properties cold and be able to derive (not just recite) its area, perimeter, and height.
Properties of an Equilateral Triangle
Three equal sides. All sides have the same length ( a ). This is the defining property.
Three equal angles. Each interior angle is 60°.
It is equiangular and regular. Equal sides and equal angles together make it a regular polygon.
Three lines of symmetry. A line from each vertex to the midpoint of the opposite side is an axis of symmetry.
The special lines coincide. For each side, the median (vertex to midpoint), the altitude (vertex perpendicular to the side), the angle bisector, and the perpendicular bisector are all the same line. In other triangles, these are usually four different lines; the equilateral triangle's symmetry collapses them into one.
It cannot be a right triangle. A natural question: can an equilateral triangle have a right angle? No. Every angle is 60°, and 60°≠90°, so there is no room for a right angle.
Equilateral Triangle Formulas
Let the side length be ( a ). Three formulas do most of the work — and each one comes from somewhere, so it is worth seeing why rather than memorising.
Perimeter
The perimeter is the total distance around the triangle. With three sides each of length ( a ):
[ P = a + a + a = 3a ]
Height (altitude)
Drop a perpendicular from one vertex to the opposite side. By symmetry, it hits the midpoint, splitting the equilateral triangle into two identical right triangles. Each right triangle has hypotenuse ( a ) (the original side) and base ( \frac{a}{2} ) (half the bottom side). Call the height ( h ) and use the Pythagorean relation:
[ h^2 + \left( \frac{a}{2} \right)^2 = a^2 ]
Solving gives:
[ h = \frac{\sqrt{3}}{2} a ]
Area
Area of any triangle is ( \frac{1}{2} \times \text{base} \times \text{height} ). Using the base ( a ) and the height we found:
[ \text{Area} = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2} a = \frac{\sqrt{3}}{4} a^2 ]
| Quantity | Formula | What the variable means |
|---|---|---|
| Perimeter | ( P = 3a ) | ( a ) is one side length |
| Height | ( h = \frac{\sqrt{3}}{2} a ) | ( a ) is one side length |
| Area | ( A = \frac{\sqrt{3}}{4} a^2 ) | ( a ) is one side length |
Examples of Equilateral Triangle
Example 1
Find the perimeter of an equilateral triangle with side 9 cm.
[ P = 3a = 3 \times 9 = 27 \text{ cm} ]
Final answer: 27 cm.
Example 2
An equilateral triangle has a perimeter of 60 cm. Find the actual side length.
[ 3a = 60 \Rightarrow a = \frac{60}{3} = 20 \text{ cm} ]
Final answer: each side is 20 cm.
Example 3
Find the area of an equilateral triangle with side 20 inches.
[ A = \frac{\sqrt{3}}{4} (20)^2 = \frac{\sqrt{3}}{4} \times 400 = 100\sqrt{3} \text{ square inches} ]
Final answer: ( 100\sqrt{3} \text{ in}^2 ) (about 173.2 in²).
Example 4
Find the height of an equilateral triangle whose side is 40 inches.
[ h = \frac{\sqrt{3}}{2} \times 40 = 20\sqrt{3} \text{ inches} ]
Final answer: ( 20\sqrt{3} \text{ inches} ) (about 34.6 inches).
Example 5
An equilateral triangle has an area of 36√3 cm². Find its side length.
Set the area formula equal to the given value:
[ \frac{\sqrt{3}}{4} a^2 = 36\sqrt{3} \Rightarrow a^2 = 144 \Rightarrow a = 12 \text{ cm} ]
Final answer: 12 cm.
Example 6
A triangular garden bed is to be built as an equilateral triangle with each side 6 m. Find both area and height.
Area:
[ A = \frac{\sqrt{3}}{4} (6)^2 = 9\sqrt{3} \approx 15.59 \text{ m}^2 ]
Height:
[ h = \frac{\sqrt{3}}{2} \times 6 = 3\sqrt{3} \approx 5.20 \text{ m} ]
Final answer: area ( \approx 15.59 \text{ m}^2 ); path ( \approx 5.20 , ext{m} ).
Why The Equilateral Triangle Matters: Maximum Strength, Maximum Symmetry
The equilateral triangle is not just the prettiest triangle. Its three-fold symmetry makes it the most efficient and stable triangle.
Structural strength. Distributes load evenly across its three sides.
Tiling and packing. Tiles a flat surface perfectly with no gaps.
The unifying idea. The first regular polygon; studying it reveals principles of regularity.
Where Students Slip With Equilateral Triangles
Mistake 1: Forgetting the 3√3 in the area and height
Mistake 2: Multiplying instead of dividing to recover a side
Mistake 3: Assuming "equilateral" allows a right or obtuse version
Key Takeaways
An equilateral triangle has three equal sides and three 60° angles.
Perimeter is 3a, height is ( \frac{\sqrt{3}}{2}a ), and area is ( \frac{\sqrt{3}}{4} a^2 ).
Its special lines coincide on each side; it is always acute.
Its symmetry makes it the strongest and most efficient triangle.