Example 1
- a = 2
h = √3 ≈ 1.732051
What is the height at a = 2? √3 ≈ 1.732051.
Equilateral Triangle Height Calculator
Enter side a. The calculator computes height h = (√3/2)·a. At a = 2 the result is √3 ≈ 1.732; at a = 6 about 5.196; at a = 10 about 8.660. The formula follows from the Pythagorean theorem on a 30-60-90 half. Domain a > 0. The primary result is height, not area — A = (√3/4)·a² is only a check. The input is side a; the calculator does not take height instead of the side.
Result
Enter side a (in). Height appears here.
h = (√3/2)·a. Area A = (√3/4)·a² is on the equilateral triangle area page. The Pythagorean theorem has its own calculator.
How it works
a2
(√3/2)·a√3
h≈ 1.732051
The height of an equilateral triangle, written h, is the segment from any vertex perpendicular to the opposite side. In this figure every side has length a and every interior angle is 60°. The altitude is also a median (it halves the base into a/2) and an angle bisector. Mathematical model: the classical equilateral-triangle height from Euclidean geometry, derived via the Pythagorean theorem.
Derivation: draw the altitude from a vertex to the base. You get two 30-60-90 right triangles with legs a/2 and h and hypotenuse a. By Pythagoras: (a/2)² + h² = a², so h² = a² − a²/4 = 3a²/4, hence h = (√3/2)·a. Equivalently h = a·sin(60°), because sin(60°) = √3/2. The calculator implements a * √(3) / 2.
Link to area: A = (1/2)·a·h. Substituting h yields A = (√3/4)·a². At a = 2: h = √3 ≈ 1.732051, A ≈ 1.732051 (different units). At a = 1: h ≈ 0.866025. At a = 6: h = 3√3 ≈ 5.196152. At a = 10: h = 5√3 ≈ 8.660254. Domain a > 0; blank and non-positive values are errors.
In an equilateral triangle the orthocenter, centroid, incenter, and circumcenter coincide. The centroid divides the altitude in a 2:1 ratio from the vertex, so it lies at distance (2/3)·h from the vertex. That is classical Euclidean geometry; Euclid’s Elements (Book I) cover altitude constructions and equal angles in an equilateral triangle.
One field: side a. Comma and period are equivalent. The unit switch only changes the cm or in label; it does not multiply the number. This is not the area calculator — the primary result is h. Area and perimeter 3a appear beside it as checks from the same a. From height back to side use a = 2h/√3.
Formula
h = (√3 / 2) · a, equivalently h = √(a² − (a/2)²).
Symbol a is the side length. Symbol h is the height (perpendicular from a vertex to the base). Half-base is a/2. Pythagorean derivation: (a/2)² + h² = a² ⇒ h = a√3/2.
Mathematical model: classical equilateral-triangle height (Euclidean geometry / Elements, Book I), via the 30-60-90 split.
Theory, 30-60-90, and area
Definition
The height (altitude) of an equilateral triangle of side a > 0 is the length of the perpendicular from a vertex to the opposite side. The canonical formula is h = (√3/2)a = a·√(3)/2. This calculator implements exactly that form, matching calculator-wysokosc-trojkata-rownobocznego.js.
The altitude bisects the base into a/2 and forms two 30°–60°–90° right triangles. By Pythagoras: h² + (a/2)² = a² ⇒ h = (√3/2)a. Area follows via A = (1/2)a h = (√3/4)a² — a companion page takes area as the primary result.
The companion panel shows half-base a/2, area, perimeter 3a, ratio h/a = √3/2, and centroid height 2h/3 from the vertex. Domain: a > 0.
Historical and conceptual context
Equilateral height is a classic Pythagoras and 60° trigonometry exercise (sin 60° = √3/2, so h = a sin 60°). MathWorld (Equilateral Triangle) and Wikipedia (Equilateral triangle) list h = (√3/2)a with R = a/√3 and r = a√3/6.
Conceptually the altitude is simultaneously a median, angle bisector, and perpendicular bisector in an equilateral triangle — they concur at the center. This page computes length h, not center coordinates.
In engineering h appears in triangular bar sections, meshes, and “rise” estimates from a known side.
Formal properties
Domain: a > 0. Codomain: h > 0. The map a ↦ h(a) = (√3/2)a is linear and strictly increasing. The ratio h/a = √3/2 ≈ 0.866025 is constant for every a.
Identities: h = (√3/2)a; A = (1/2)a h = (√3/4)a²; R = 2h/3; r = h/3; half-base = a/2; centroid: 2h/3 from the vertex, h/3 from the base.
Homogeneity: h(λa) = λ h(a). For a = 2/√3 the height equals 1.
Height h is also a median and an angle bisector: M is the midpoint, and the angles at A split as 30°+30°. Coordinates with base (−a/2, 0), (a/2, 0) and apex (0, h) are the canonical embedding of the equilateral triangle.
The ratio h/a = √3/2 ≈ 0.866025 is the dimensionless similarity constant of all equilateral triangles. The panel prints that ratio beside h to emphasize linear homogeneity.
Compared with the area calculator: this page takes h as primary, that page takes A = (√3/4)a². Both share √3 and the same half-triangle Pythagoras; do not confuse the coefficients (√3/2 vs √3/4) when copying formulas.
Trigonometry: h = a·sin 60° = a·cos 30°, both equal to (√3/2)a. The apothem of a regular hexagon of side a equals the same value — a bridge between the equilateral triangle and regular polygons.
Full derivation
Step 1. In equilateral ABC drop the altitude from A to BC at M. Then BM = a/2, ∠AMB = 90°, ∠ABM = 60°.
Step 2. Pythagoras: h² + (a/2)² = a² ⇒ h = (√3/2)a.
Step 3. Trigonometry: h = a sin 60° = a·(√3/2) — the same result.
Step 4. Area: A = (1/2)a h confirms consistency with A = (√3/4)a².
Numeric checks: a = 2 ⇒ h = √3; a = 1 ⇒ h = √3/2; a = 6 ⇒ h = 3√3; a = 10 ⇒ h = 5√3. Consistency with A = (1/2)a h on the area companion and with R = 2h/3, r = h/3 on the panel confirms the coefficient √3/2 was not confused with √3/4. The SVG altitude is a drawing aid only and does not change the numeric height.
Special cases
a = 2: h = √3 ≈ 1.732051. a = 1: h = √3/2 ≈ 0.866025. a = 6: h = 3√3 ≈ 5.196152. a = 10: h = 5√3 ≈ 8.660254.
As a → 0⁺, h → 0 linearly. Doubling a doubles h (degree 1), while area grows fourfold.
In a regular hexagon of side a the apothem equals (√3/2)a — the same constant as h in an equilateral triangle of side a.
Numerical and computational notes
Computation: SQRT3 = √(3); h = a·SQRT3/2. Input a ≤ 0 is rejected. √3 is an approximate constant sufficient for displayed decimals.
Examples and the diagram refresh on input. A comma and a period are equivalent; the unit switch changes the length-unit label, not the √3/2 coefficient.
Circle relations: circumcenter and incenter coincide with the centroid; R = 2h/3, r = h/3, so h = R + r. The identity h = 3r is handy in problems that give height alone without naming a.
If a is in cm then h is in cm; the US toggle changes the label to in without an extra numeric conversion beyond shared Format.getUnits. The calculator rejects a ≤ 0 with a domain message.
Worked values a ∈ {1, 2, 6, 10} mirror the area companion so cross-checking h and A = (1/2)a h is immediate for students and auditors.
Related concepts and authorities
Related: equilateral area A = (√3/4)a², angle sum (60°), Pythagoras. This page takes h as the primary result.
Authorities: MathWorld — Equilateral Triangle; Wikipedia — Equilateral triangle; CRC / Zwillinger. The calculator reports h = (√3/2)a.
How to use
- Enter a positive side a, for example 2. Use a comma or a period.
- Click Calculate or wait for the live result. a = 2 gives h ≈ 1.732051.
- Check the panel: h, a/2, area, perimeter, ratio h/a, (2/3)·h, and the altitude diagram.
- Paint area on the triangle lives on the equilateral-area page. Pythagoras has its own page.
- The unit switch only changes the cm↔in label; it does not convert the number.
Symbol glossary
h = (√3/2)·a. At a = 2, height ≈ 1.732051. Domain: a > 0.
- a
- Side of the equilateral triangle. Input. Must be > 0. At a = 2: h = √3.
- h
- Height — primary result. h = (√3/2)·a ≈ 0.866025·a.
- a/2
- Half-base after drawing the altitude. Leg in the 30-60-90 triangle.
- A
- Area (√3/4)·a² = (1/2)·a·h. Companion check, not this page’s primary result.
Examples
Example 2
- a = 1
h = √3/2 ≈ 0.866025
What is the height at a = 1? √3/2 ≈ 0.866025.
Example 3
- a = 6
h = 3√3 ≈ 5.196152
What is the height at a = 6? 3√3 ≈ 5.196152.
Example 4
- a = 10
h = 5√3 ≈ 8.660254
What is the height at a = 10? 5√3 ≈ 8.660254.
Related calculators
Common questions
What is the height of an equilateral triangle?
Height h is the segment from a vertex perpendicular to the opposite side (the base). In an equilateral triangle that altitude is also a median and an angle bisector. Formula: h = (√3/2)·a.
Where does h = (√3/2)·a come from?
The altitude splits the triangle into two 30-60-90 right triangles: half-base a/2, hypotenuse a, other leg h. By the Pythagorean theorem: h = √(a² − (a/2)²) = √(3a²/4) = (√3/2)·a.
How does height relate to area?
Triangle area is A = (1/2)·a·h. Substituting h = (√3/2)·a yields A = (√3/4)·a². At a = 2 in, height and area share the numeric value √3 ≈ 1.732, but different units (in vs in²).
Why does the domain require a > 0?
A side length describes a real triangle only for positive values. Zero is not a figure; a negative length is not a length. A blank field is an error, not height 0.
Is this the same as the equilateral area calculator?
No. Here the primary result is height h from one side a. Area A = (√3/4)·a² is a companion check and has its own equilateral-area page.
Do altitude, median, and orthocenter coincide?
Yes. In an equilateral triangle the altitudes, medians, angle bisectors, and perpendicular bisectors meet at one point. Orthocenter, centroid, incenter, and circumcenter lie on the same altitude.
Do the currency and unit settings in the page header affect the result?
No. The unit switch only changes the cm or in label; it does not multiply the entered digits. If you measured 2 cm and want inches, convert the length yourself first.
What is the height at a = 2?
h = (√3/2)·2 = √3 ≈ 1.732051 in the same unit as a. That is the first example on this page. Half-base is 1; companion area ≈ 1.732051 square units.
Can I enter height instead of side?
No. The input is side a. From height h you recover the side with a = 2h/√3 = (2√3/3)·h, then you can return here with that a.
Where is the centroid on the altitude?
The centroid divides each median (here: the altitude) in a 2:1 ratio from the vertex. Distance from the vertex is (2/3)·h; from the base (1/3)·h. At a = 2 that is about 1.155 and 0.577.
How do I calculate the height of an equilateral triangle?
Multiply side a by √3/2: h = (√3/2)·a. At a = 2 you get √3 ≈ 1.732; at a = 6 about 5.196. This is not the height formula for a general triangle from three sides.
What is the height at a = 6?
3√3 ≈ 5.196152 in the same unit as a. a/2 = 3. That is the third example on this page.
What is the height at a = 10?
5√3 ≈ 8.660254. Perimeter is 30; the centroid lies at (2/3)·h ≈ 5.773503 from the vertex. Fourth example.
What is the height at a = 8?
4√3 ≈ 6.928203. Same formula: h = (√3/2)·8. That side has no separate example card, but the calculator accepts it.
Is this the height of any triangle?
No. Here every side equals a and every angle is 60°. Height of a general triangle from three unequal sides or from area needs a different formula and a different calculator.
Knowledge sources
The formula h = (√3/2)·a, the 30-60-90 split, and the Elements note below come from recognized sources.
- MathWorld — Equilateral Triangle
- Wikipedia — Equilateral triangle
- Wikipedia — Pythagorean theorem
- Wikipedia — 30-60-90 triangle
- Wikipedia — Euclid's Elements
- MathWorld — Altitude
Page updated November 3, 2026.