How to Calculate the Area of an Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal and each interior angle measures $60^\circ$. The computation of its area relies on geometric properties that uniquely arise due to this symmetry. The area formula for an equilateral triangle expresses the region enclosed in terms of the side length, and can be derived from elementary principles.

Area Formula for Equilateral Triangle in Terms of Side Length

Let the side length of an equilateral triangle be denoted by $a$. The general area formula for any triangle using base and height is $A = \dfrac{1}{2} \times$ base $\times$ height. For an equilateral triangle, take base $= a$.

To use this formula, express the altitude (height) in terms of $a$. Place the triangle with base $BC$ on the $x$-axis, and vertex $A$ above $BC$. Draw the altitude from $A$ to $D$ on $BC$. Since the triangle is equilateral, $BD = DC = \dfrac{a}{2}$, and $AB = AC = BC = a$.

Apply the Pythagorean theorem to triangle $ABD$: \[ AB^2 = AD^2 + BD^2 \] \[ a^2 = h^2 + \left(\dfrac{a}{2}\right)^2 \]

Evaluate $\left(\dfrac{a}{2}\right)^2$: \[ \left(\dfrac{a}{2}\right)^2 = \dfrac{a^2}{4} \] Therefore, \[ a^2 = h^2 + \dfrac{a^2}{4} \]

Subtract $\dfrac{a^2}{4}$ from both sides: \[ a^2 - \dfrac{a^2}{4} = h^2 \] \[ \dfrac{4a^2}{4} - \dfrac{a^2}{4} = h^2 \] \[ \dfrac{3a^2}{4} = h^2 \]

Take the square root of both sides: \[ h = \dfrac{\sqrt{3}}{2}a \]

Substitute back into the general triangle area formula: \[ A = \dfrac{1}{2} \times a \times \dfrac{\sqrt{3}}{2}a \] \[ A = \dfrac{1}{2} \times \dfrac{\sqrt{3}}{2} a^2 \] \[ A = \dfrac{\sqrt{3}}{4} a^2 \]

Result: The area of an equilateral triangle with side $a$ is $A = \dfrac{\sqrt{3}}{4}a^2$.

Derivation Using Heron's Formula for Area of Equilateral Triangle

Heron's formula gives the area of a triangle when all three side lengths are known. For a triangle with sides $a$, $b$, $c$, \[ A = \sqrt{s(s - a)(s - b)(s - c)} \] where $s$ is the semi-perimeter, \[ s = \dfrac{a + b + c}{2} \]

In an equilateral triangle, $a = b = c$, so the semi-perimeter is: \[ s = \dfrac{3a}{2} \]

Substitute $a = b = c$ into Heron's formula: \[ A = \sqrt{s(s - a)^3} \]

Now compute $s - a$: \[ s - a = \dfrac{3a}{2} - a = \dfrac{3a - 2a}{2} = \dfrac{a}{2} \]

Substitute the expressions: \[ A = \sqrt{s \cdot (s - a)^3} = \sqrt{ \dfrac{3a}{2} \cdot \left(\dfrac{a}{2}\right)^3 } \]

Compute $\left(\dfrac{a}{2}\right)^3$: \[ \left(\dfrac{a}{2}\right)^3 = \dfrac{a^3}{8} \]

So, \[ A = \sqrt{ \dfrac{3a}{2} \cdot \dfrac{a^3}{8} } \] \[ A = \sqrt{ \dfrac{3a^4}{16} } \] \]

Take the square root of the fraction and numerator: \[ A = \dfrac{\sqrt{3a^4}}{4} \] \[ A = \dfrac{\sqrt{3} \cdot a^2}{4} \]

Result: Using Heron's formula, $A = \dfrac{\sqrt{3}}{4} a^2$ for an equilateral triangle.

Area of Equilateral Triangle in Terms of Height

Given the altitude $h$ of an equilateral triangle, express $a$ in terms of $h$ using the previously obtained relation: \[ h = \dfrac{\sqrt{3}}{2}a \] \[ a = \dfrac{2h}{\sqrt{3}} \]

Substitute $a = \dfrac{2h}{\sqrt{3}}$ in the area formula: \[ A = \dfrac{\sqrt{3}}{4}a^2 \] \[ = \dfrac{\sqrt{3}}{4} \left( \dfrac{2h}{\sqrt{3}} \right)^2 \] \[ = \dfrac{\sqrt{3}}{4} \cdot \dfrac{4h^2}{3} \] \[ = \dfrac{\sqrt{3} \cdot 4h^2}{4 \cdot 3} \] \[ = \dfrac{h^2}{\sqrt{3}} \]

Result: The area in terms of height $h$ is $A = \dfrac{h^2}{\sqrt{3}}$.

Area of Equilateral Triangle Using Trigonometry

For any triangle with two sides $b$ and $c$, and included angle $\theta$, the area is $A = \dfrac{1}{2} bc \sin\theta$. In an equilateral triangle, take $b = c = a$, and $\theta = 60^\circ$, so \[ A = \dfrac{1}{2} a \cdot a \cdot \sin 60^\circ \] \[ = \dfrac{1}{2} a^2 \cdot \dfrac{\sqrt{3}}{2} \] \[ = \dfrac{\sqrt{3}}{4} a^2 \]

Result: The area formula remains $A = \dfrac{\sqrt{3}}{4} a^2$.

Explicit Worked Examples for Area of Equilateral Triangle

Example 1: Find the area of an equilateral triangle of side $6$ cm.

Given $a = 6$ cm.
Substitute in the formula: \[ A = \dfrac{\sqrt{3}}{4} a^2 \] \[ = \dfrac{\sqrt{3}}{4} \times (6)^2 \] \[ = \dfrac{\sqrt{3}}{4} \times 36 \] \[ = 9\sqrt{3} \text{ cm}^2 \]

Example 2: The altitude of an equilateral triangle is $12\ \text{cm}$. Find its area.

Given $h = 12$ cm.
Use area in terms of height: \[ A = \dfrac{h^2}{\sqrt{3}} \] \[ = \dfrac{(12)^2}{\sqrt{3}} \] \[ = \dfrac{144}{\sqrt{3}} \] Rationalizing denominator: \[ = \dfrac{144\sqrt{3}}{3} = 48\sqrt{3} \text{ cm}^2 \]

Example 3: The perimeter of an equilateral triangle is $18$ units. Find its area.

Given perimeter $= 3a = 18$ units, so $a = \dfrac{18}{3} = 6$ units.
Substitute in area formula, \[ A = \dfrac{\sqrt{3}}{4} (6)^2 = \dfrac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3}\ \text{units}^2 \]

Area Of Equilateral Triangle Formula and alternate triangle area results can be found for comparison and practice.

Area of Equilateral Triangle: Relevant Formulas and Further Results

Side in terms of Area: If area $A$ is known, \[ A = \dfrac{\sqrt{3}}{4} a^2 \] \[ a^2 = \dfrac{4A}{\sqrt{3}} \] \[ a = \sqrt{ \dfrac{4A}{\sqrt{3}} } \]

Perimeter in terms of Area: \[ a = \sqrt{ \dfrac{4A}{\sqrt{3}} } \] So perimeter $= 3a = 3 \sqrt{ \dfrac{4A}{\sqrt{3}} }$

For related formulas on polygons and area, see Area Of Square Formula, Area Of Isosceles Triangle Formula, and Area Formula For Quadrilateral.