Example 1
- a = 2
A ≈ 1.732051 · h ≈ 1.732051 · P = 6
What is the equilateral triangle area at a = 2? ≈ 1.732051.
Equilateral triangle area calculator
Enter side length a (a > 0). Area is A = (√3/4)·a². With a = 2 the result is √3 ≈ 1.732051. With a = 6 the result is 9√3 ≈ 15.588457. The panel also shows height h = (√3/2)·a and perimeter 3a. This calculator accepts only the side — not height or perimeter as inputs. For a scalene triangle use Heron’s formula or ½·base·height on the other pages.
Result
Enter side length a.
One side is enough because all three match. Domain: a > 0. This is not area from a separate base and height — here the factor √3/4 replaces the second dimension.
How it works
a2
h√3
(√3/4)·a²√3
A≈ 1.732051
An equilateral triangle has three equal sides and three 60° angles. One side length a is enough to compute area, height, and perimeter. This is not a generic triangle with base and height entered separately — every side matches, so the formula collapses to the factor √3/4.
The classic formula is A = (√3/4)·a². Equivalently A = ½·a·h with h = (√3/2)·a. Derivation: the altitude to the base splits the triangle into two 30°–60°–90° right triangles with hypotenuse a and base a/2; Pythagoras gives h = √(a² − a²/4) = (√3/2)·a, so the area is ½·a·h. Heron’s formula on sides a, a, a yields the same result: s = 3a/2 and √[s(s−a)³] = (√3/4)·a². Constructing an equilateral triangle is Euclid’s Elements Book I, Proposition 1; the area formula is standard Euclidean geometry in school and university texts.
The calculator uses the number you enter: side length a. Domain: a > 0. At a = 2 the area is √3 ≈ 1.732051, height is also ≈ 1.732051, perimeter 6. At a = 6 the area is 9√3 ≈ 15.588457, height 3√3 ≈ 5.196152, perimeter 18. At a = 10 the area is 25√3 ≈ 43.301270. At a = 1 the area is √3/4 ≈ 0.433013.
The neighboring equilateral-height page computes only h = (√3/2)·a; the generic triangle-area page takes any base and height.
Formula
A = (√3/4)·a², where a > 0 is the side length.
Height h = (√3/2)·a. Perimeter P = 3a. Equivalently A = ½·a·h.
Heron for sides a, a, a: s = 3a/2, A = √[s(s−a)³] — simplifies to (√3/4)·a².
Mathematical model: equilateral triangle area — standard Euclidean geometry (Euclid, Elements I.1; textbook A = (√3/4)a²).
Theory, derivation, and domain
Definition
An equilateral triangle has all three sides of equal length a > 0 and all three interior angles equal to 60°. Its area is A = (√3/4)a². This calculator computes exactly A = (√(3)/4)·a·a together with height h = (√3/2)a and perimeter P = 3a.
The formula follows from the altitude to the base: the altitude splits the figure into two 30°–60°–90° right triangles with legs a/2 and h = a·√3/2. Hence A = (1/2)·a·h = (1/2)·a·(a√3/2) = (√3/4)a². Heron’s formula on sides a, a, a yields the same area: semiperimeter s = 3a/2, A = √[s(s−a)³].
The area unit is the square of the side unit (cm² or in² via the unit switch). The primary result is area A; height and perimeter appear beside it.
Historical and conceptual context
The equilateral triangle is a classical Euclidean figure (Elements I.1 constructs an equilateral triangle on a given segment). The area A = (√3/4)a² is standard in school and engineering; MathWorld (Equilateral Triangle) and Wikipedia (Equilateral triangle) list area, height, inradius, circumradius, and the √3 factor.
Conceptually “equilateral” means equal sides; in Euclidean geometry equal angles (60°) follow. The constant √3 comes from Pythagoras in the half-triangle: h² + (a/2)² = a² ⇒ h = (√3/2)a.
In practice the formula serves triangular meshes, bar cross-sections, tiling, and area estimates from a single side length. This page does not solve a scalene triangle — use the laws of sines and cosines for that.
Formal properties
Domain: a > 0. Codomain: A > 0. The map a ↦ A(a) = (√3/4)a² is strictly increasing and homogeneous of degree 2. Height h = (√3/2)a is linear in a; the ratio h/a = √3/2 is constant. Interior angle α = 60° = π/3 for every a.
Identities: A = (√3/4)a²; h = (√3/2)a; P = 3a; circumradius R = a/√3; inradius r = a√3/6; A = (1/2)a h. From Heron: s = 3a/2, A = √[s(s−a)³] = (√3/4)a².
The centroid lies on the altitude at distance 2h/3 from the vertex (or h/3 from the base). Circumradius R = 2h/3; inradius r = h/3.
The formula A = (√3/4)a² equals (1/2)a² sin 60° from the general triangle area law with a = b and included angle 60°, since (1/2)a²(√3/2) = (√3/4)a². Circumradius R = a/√3 and inradius r = a√3/6 satisfy R = 2r, and the altitude obeys h = 3r = (3/2)R in the usual concurrence geometry of the equilateral triangle.
In a regular triangular tiling of the plane each cell has area (√3/4)a²; cell density per unit area is 1/A = 4/(√3 a²). Those facts link the area calculator to discrete geometry without changing A itself.
Full derivation
Step 1. Let ABC be equilateral with side a. Drop the altitude from A to BC at M. Then BM = MC = a/2 and ∠AMB = 90°.
Step 2. In right triangle AMB: AM² + (a/2)² = a², so AM² = (3/4)a² and h = AM = (√3/2)a (positive branch).
Step 3. Area: A = (1/2)·base·height = (1/2)·a·h = (√3/4)a².
Step 4 (Heron). s = 3a/2, s−a = a/2. A = √[s(s−a)³] = √[(3a/2)(a/2)³] = (√3/4)a².
Special cases
a = 1: A = √3/4 ≈ 0.433013; h = √3/2 ≈ 0.866025. a = 2: A = √3 ≈ 1.732051; h = √3. a = 6: A = 9√3 ≈ 15.588457; h = 3√3 ≈ 5.196152. a = 10: A = 25√3 ≈ 43.301270; h = 5√3 ≈ 8.660254.
As a → 0⁺, A → 0. Doubling the side multiplies area by four (degree-2 homogeneity). Six equilateral triangles of side a tile a regular hexagon of area (3√3/2)a².
The 60° angle links this model to the regular hexagon and to the n = 3 case of the polygon interior-angle sum (S = 180°, regular angle 60°).
Numerical and computational notes
The calculator evaluates √3 once and computes A = (√3/4)·a² and h = (√3/2)·a. An on-page Heron check matches the floating-point result for ordinary a. Input a ≤ 0 is rejected.
Comma and period are equivalent. Examples fill a ∈ {1, 2, 6, 10}. The SVG diagram scales the side to a fixed view width; drawing scale does not change the numeric result.
Numerically the page also evaluates Heron’s √[s(s−a)³] with s = 3a/2; the gap versus (√3/4)a² in floating-point arithmetic for school-scale a is rounding error, not a formula error. The calculator rejects a ≤ 0 and does not treat a negative “side” as a reflected figure.
Related lengths on the panel (height, perimeter) are derived from the same a; they do not feed back into A. US/Metric only switches the displayed unit label for a and A.
Related concepts and authorities
Related: equilateral height h = (√3/2)a (companion page), polygon angle sum at n = 3, Pythagoras in the half-triangle.
Authorities: MathWorld — Equilateral Triangle; Wikipedia — Equilateral triangle; Euclid, Elements I.1; CRC / Zwillinger triangle formulas. The calculator reports A = (√3/4)a².
How to use
- Enter side length a greater than zero, for example 2 or 6.
- Click Calculate or wait for the live result. At a = 2 the area is ≈ 1.732051.
- Check the panel: height, perimeter, half-base, Heron check, and the altitude diagram.
- Header units change the cm² / in² label; they do not convert the entered number.
- Area from any base and height lives on the triangle area page; height alone on equilateral triangle height.
Symbol glossary
At a = 2 the area is √3 ≈ 1.732051. Height at the same a is also √3.
- A
- Equilateral area: (√3/4)·a². At a = 2 it is ≈ 1.732051.
- a
- Side length in the form field. Must be greater than zero.
- h
- Height: (√3/2)·a. At a = 2 ≈ 1.732051; at a = 6 ≈ 5.196152.
- P
- Perimeter 3a. At a = 2 it is 6; at a = 6 it is 18.
- s
- Semiperimeter 3a/2 used in the Heron check.
- √3
- Constant from the 30°–60°–90° height. √3 ≈ 1.732051.
Examples
Example 2
- a = 6
A ≈ 15.588457 · h ≈ 5.196152 · P = 18
What is the area at a = 6? ≈ 15.588457.
Example 3
- a = 10
A ≈ 43.301270 · h ≈ 8.660254 · P = 30
What is the area at a = 10? ≈ 43.301270.
Example 4
- a = 1
A ≈ 0.433013 · h ≈ 0.866025 · P = 3
What is the area at a = 1? ≈ 0.433013.
Related calculators
Frequently asked questions
What formula computes equilateral triangle area?
The area is A = (√3/4)·a², where a is the side length and a > 0. Equivalently A = ½·a·h with height h = (√3/2)·a. That matches Heron’s formula when all three sides equal a.
Where does √3 come from in the formula?
The altitude splits the triangle into two 30°–60°–90° right triangles with hypotenuse a and base a/2. Pythagoras gives h = √(a² − (a/2)²) = (√3/2)·a. Then A = ½·a·h = (√3/4)·a².
What is the domain for the side length?
The side must be greater than zero (a > 0). When a ≤ 0 there is no triangle with positive area — the calculator shows a message instead of a false result.
How do height and perimeter follow from the same a?
Height h = (√3/2)·a and perimeter P = 3a. Altitude, median, and angle bisector coincide in an equilateral triangle. From h you can recover a = 2h/√3 — this calculator starts from a, not from h.
What is the area at a = 2 and at a = 6?
At a = 2 the area is √3 ≈ 1.732051, height is also ≈ 1.732051, perimeter 6. At a = 6 the area is 9√3 ≈ 15.588457, height 3√3 ≈ 5.196152, perimeter 18. Those are examples 1 and 2 on this page.
How does this page differ from base-and-height triangle area?
The neighboring triangle-area calculator takes any base and a perpendicular height. Here every side is equal, so one side a and the factor √3/4 are enough. A 3-4-5 triangle is not equilateral.
Does Heron’s formula give the same area?
Yes. For sides a, a, a the semiperimeter s = 3a/2, and √[s(s−a)³] simplifies to (√3/4)·a². The result panel also shows a Heron check beside the √3 formula.
Can I enter height instead of the side?
Not in this form — the input is side a. From height h first compute a = 2h/√3 (or use the equilateral height calculator in reverse), then come back here.
Do the currency and unit settings in the page header affect the result?
No. The result depends only on a. The header attaches an in or cm label and in² or cm². An entered 2 stays 2 — it is not converted to inches. The currency and unit settings in the page header do not affect this result.
Does a comma in 2.5 work?
Yes. 2.5 and 2,5 are the same side length. A comma or a period in the numeric field is accepted.
How does the result differ from a square with the same side?
A square of side a has area a². An equilateral triangle has (√3/4)·a² ≈ 0.433·a² — less than half the square. At a = 2 the square has area 4, while the triangle has √3 ≈ 1.732.
What else does the result panel show?
Besides area A you see height h, perimeter 3a, half-base a/2, semiperimeter s, a Heron check, a², and √3. The live diagram shows the triangle with its altitude.
How do I calculate the area of an equilateral triangle?
Use A = (√3/4)·a², where a is the side length. With a = 2: (√3/4)·4 = √3 ≈ 1.732051. Equivalently A = ½·a·h with h = (√3/2)·a.
What is the formula for equilateral triangle area?
A = (a²·√3)/4. With a = 10: 100·√3/4 = 25√3 ≈ 43.301270. Area scales with the square of the side — doubling a multiplies A by four.
What is the area of an equilateral triangle with side 5?
≈ 10.825318. Enter a = 5: A = 25√3/4. Height is then (5√3)/2 ≈ 4.330127, and the perimeter is 15.
Does Heron's formula give the same result for three equal sides?
Yes. With a = b = c = 6: s = 9, A = √(9·3·3·3) = √243 = 9√3 ≈ 15.588457 — the same as (√3/4)·36. Here one side field is enough instead of three.
Sources
The formula, domain, and derivation below come from established geometry sources.
- MathWorld — Equilateral Triangle
- Wikipedia — Equilateral triangle
- Euclid, Elements, Book I, Proposition 1 (equilateral construction).
- Hartshorne, R., Geometry: Euclid and Beyond, Springer (Euclidean geometry and areas).
Page updated November 3, 2026.