What Is an Equilateral Triangle?
An equilateral triangle, also called a regular triangle, is a triangle in which all three sides are equal in length. Because of this, all three interior angles are also equal — each measuring exactly 60°. This makes it the most symmetric of all triangles.
Key properties of an equilateral triangle:
- All three sides are equal: a = b = c
- All three angles are equal: α = β = γ = 60°
- The altitude, median, angle bisector, and perpendicular bisector from each vertex all coincide
- It is a special case of the isosceles triangle (with all sides equal instead of just two)
Equilateral Triangle Area and Height
The area formula for an equilateral triangle with side length a is:
A = (a² × √3) / 4
The height (altitude) from any vertex to the opposite side is:
h = a × √3 / 2
Derivation Using the Pythagorean Theorem
Drop a perpendicular from the top vertex to the base. This splits the equilateral triangle into two congruent right triangles. Each right triangle has:
- Hypotenuse = a (the original side)
- One leg = a / 2 (half the base)
- Other leg = h (the height)
By the Pythagorean theorem: (a/2)² + h² = a² → h² = a² − a²/4 = 3a²/4 → h = a√3 / 2
Substituting into the base triangle area formula (base × height / 2):
A = a × (a√3/2) / 2 = a²√3 / 4
Derivation Using Trigonometry
Using the general triangle area formula A = (1/2) × a × b × sin(γ), and knowing all sides are a and all angles are 60°:
A = (1/2) × a × a × sin(60°) = (1/2) × a² × (√3/2) = a²√3 / 4
The height follows directly from the sine definition: h / a = sin(60°), so h = a × sin(60°) = a√3 / 2.
Perimeter, Circumcircle, and Incircle
Since all sides are equal, the perimeter is simply:
P = 3 × a
The circumscribed circle (circumcircle) passes through all three vertices. Its radius is:
R = 2h / 3 = a√3 / 3 = a / √3
The inscribed circle (incircle) touches all three sides from the inside. Its radius is:
r = h / 3 = a√3 / 6
Note that R = 2r for an equilateral triangle — the circumcircle is always twice as large as the incircle.
How to Use This Equilateral Triangle Calculator
- Select your preferred unit system — Metric (mm, cm, m, km) or Imperial (in, ft, yd, mi).
- Choose the specific unit from the dropdown (e.g., centimeters or inches).
- Enter the side length (a) of the equilateral triangle.
- Click Calculate. The calculator instantly returns the height, area, perimeter, circumcircle radius, and incircle radius.
Example (yield sign): A standard US yield sign has a side of 36 inches. The calculator gives:
- Height ≈ 31.18 in
- Area ≈ 561.18 in²
- Perimeter = 108 in
- Circumcircle radius ≈ 20.78 in
- Incircle radius ≈ 10.39 in
Frequently Asked Questions
How do I find the area of an equilateral triangle?
Take the square root of 3 (≈ 1.732), multiply it by the square of the side length, then divide by 4: A = a² × √3 / 4. For example, if the side is 8 cm, the area is 64 × 1.732 / 4 ≈ 27.71 cm².
How do I find the height of an equilateral triangle?
Multiply the side length by √3 (≈ 1.732) and divide by 2: h = a × √3 / 2. For a side of 8 cm, the height ≈ 8 × 1.732 / 2 ≈ 6.93 cm.
How do I find the perimeter of an equilateral triangle with a side of 8 cm?
Simply multiply the side by 3: P = 3 × 8 = 24 cm.
What is the circumcircle radius of an equilateral triangle?
The circumcircle radius (circumradius) equals R = a√3 / 3. It is exactly two-thirds of the triangle's height.
What is the incircle radius of an equilateral triangle?
The incircle radius (inradius) equals r = a√3 / 6. It is exactly one-third of the triangle's height, or half the circumradius.
Can I enter the side in different units?
Yes! This calculator supports both metric units (millimeters, centimeters, meters, kilometers) and US imperial units (inches, feet, yards, miles). Simply select your preferred unit before entering the side length.