Why is the area formula (sqrt(3)/4)s squared and not something simpler?

An equilateral triangle's height is (sqrt(3)/2)s, derived from the Pythagorean theorem applied to the two halves. Area = (1/2) x base x height = (1/2) x s x (sqrt(3)/2)s = (sqrt(3)/4)s squared.

All sides are equal — does that mean the triangle must also be equiangular?

Yes. Any triangle with all three sides equal (equilateral) must also have all three interior angles equal to 60 degrees (equiangular). The two properties are equivalent for triangles.

How do I find the circumradius or inradius of an equilateral triangle?

Circumradius R = s / sqrt(3) and inradius r = s / (2*sqrt(3)). These are not computed here, but both depend only on the side length s.

Equilateral Triangle Calculator

Calculate the area, height, and perimeter of an equilateral triangle from the side length. All three sides are equal and all angles are 60 degrees. Enter any side length to get results instantly.

By AbraCalc Editorial Team · Reviewed by AbraCalc Methodology Review · Last reviewed: 2026-06-25

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How to use this tool

Enter side length in the fields above.
Results update instantly as you type — or click Calculate.
Read your area and the full breakdown beneath it.

Formula

Area = (sqrt(3)/4)*s^2; Height = (sqrt(3)/2)*s; Perimeter = 3s

How it works

For side s: Area = (√3/4)s², Height = (√3/2)s, Perimeter = 3s.

Worked example

Side = 6

A
r
e
a
=
s
q
r
t
(
3
)
/
4
*
3
6
=
1
5
.
5
8
8
.
H
e
i
g
h
t
=
s
q
r
t
(
3
)
/
2
*
6
=
5
.
1
9
6
.
P
e
r
i
m
e
t
e
r
=
1
8
.

Common mistakes to avoid

Using the perimeter formula (3s) when the question asks for area — these are completely different outputs from the same side length.
Forgetting that the height formula gives the perpendicular height from base to opposite vertex, not the slant or any other measurement.
Squaring the side after multiplying: writing sqrt(3)/4 x s then squaring s, rather than computing s squared first and then multiplying by sqrt(3)/4.

Key terms

Frequently asked questions

Why is the area formula (sqrt(3)/4)s squared and not something simpler?: An equilateral triangle's height is (sqrt(3)/2)s, derived from the Pythagorean theorem applied to the two halves. Area = (1/2) x base x height = (1/2) x s x (sqrt(3)/2)s = (sqrt(3)/4)s squared.
All sides are equal — does that mean the triangle must also be equiangular?: Yes. Any triangle with all three sides equal (equilateral) must also have all three interior angles equal to 60 degrees (equiangular). The two properties are equivalent for triangles.
How do I find the circumradius or inradius of an equilateral triangle?: Circumradius R = s / sqrt(3) and inradius r = s / (2*sqrt(3)). These are not computed here, but both depend only on the side length s.