What Is the Height of an Equilateral Triangle?
The height of an equilateral triangle is the perpendicular distance from any one vertex to the opposite side. Because an equilateral triangle has all three sides equal and all three angles equal to 60°, this perpendicular — the altitude of a triangle — behaves specially: it lands exactly on the midpoint of the opposite side, so it is also the median of a triangle to that side. Altitude, median, perpendicular bisector, and angle bisector all coincide in an equilateral triangle.
An equilateral triangle is one of the three angle-and-side families covered in types of triangle; it is also a special case of the isosceles triangle, where here all three sides match rather than just two.
The Height of Equilateral Triangle Formula
For an equilateral triangle of side length $a$:
$$h = \frac{\sqrt{3}}{2},a$$
Variable glossary:
$h$ — the height (altitude) of the triangle
$a$ — the length of one side
$\sqrt{3} \approx 1.732$ — a fixed constant that appears because of the 60° angles
So the height is always a touch under 87% of the side length ($\sqrt{3}/2 \approx 0.866$).
Where the Formula Comes From (Derivation)
The formula is not something to memorise cold — it falls out of one application of Pythagoras. Drawing the altitude $h$ to the base creates a right triangle with:
hypotenuse $a$ (a full side),
one leg $h$ (the height),
one leg $\dfrac{a}{2}$ (half the base, because the altitude bisects it).
Apply the Pythagoras theorem to that right triangle:
$$a^2 = h^2 + \left(\frac{a}{2}\right)^2$$
Isolate $h^2$:
$$h^2 = a^2 - \frac{a^2}{4}$$
Combine the right side over a common denominator:
$$h^2 = \frac{4a^2 - a^2}{4} = \frac{3a^2}{4}$$
Take the positive square root:
$$h = \frac{\sqrt{3},a}{2}$$
That is the formula. Every other version below is this one rearranged.
Finding the Height from the Area or Perimeter
A short opener: sometimes you are not handed the side directly. Two quick conversions cover those cases.
From the perimeter $P$. Since $P = 3a$, the side is $a = \dfrac{P}{3}$. Substitute into the height formula: $h = \dfrac{\sqrt{3}}{2}\cdot\dfrac{P}{3} = \dfrac{\sqrt{3},P}{6}$.
From the area $A$. The area of an equilateral triangle is $A = \dfrac{\sqrt{3}}{4}a^2$. Solve for $a$ first, then apply the height formula — or use the direct relation $h = \dfrac{2A}{a}$ once the side is known.
In every case, find the side $a$ first, then run the one height formula. There is no need to memorise three separate formulas.
Examples of Height of Equilateral Triangle
Example 1
Find the height of an equilateral triangle with side 6 cm.
$$h = \frac{\sqrt{3}}{2},a$$
Substitute $a = 6$:
$$h = \frac{\sqrt{3}}{2}\times 6$$
$$h = 3\sqrt{3} \approx 5.20 \text{ cm}$$
Final answer: $3\sqrt{3}$ cm, about 5.20 cm.
Example 2 (a tempting shortcut that fails)
Find the height of an equilateral triangle with side 10 cm.
Wrong attempt. A common reflex is to treat the height as half the side — "the altitude cuts the triangle in half, so $h = 10/2 = 5$ cm." That confuses cutting the base in half with halving the height.
Why it breaks. The altitude bisects the base (giving the $a/2$ leg), but the height itself is the other leg of the right triangle, found through Pythagoras — not by halving anything.
Correct.
$$h = \frac{\sqrt{3}}{2}\times 10 = 5\sqrt{3} \approx 8.66 \text{ cm}$$
Final answer: $5\sqrt{3} \approx 8.66$ cm — noticeably taller than the wrong 5 cm, because the height is about 0.866 of the side, not 0.5.
Example 3
An equilateral triangle has a perimeter of 18 cm. Find its height.
First find the side from the perimeter:
$$a = \frac{P}{3} = \frac{18}{3} = 6 \text{ cm}$$
Then apply the height formula:
$$h = \frac{\sqrt{3}}{2}\times 6 = 3\sqrt{3} \approx 5.20 \text{ cm}$$
Final answer: $3\sqrt{3} \approx 5.20$ cm.
Example 4
The height of an equilateral triangle is $4\sqrt{3}$ cm. Find the length of its side.
Start from the formula and solve for $a$:
$$h = \frac{\sqrt{3}}{2},a$$
$$4\sqrt{3} = \frac{\sqrt{3}}{2},a$$
Multiply both sides by $\dfrac{2}{\sqrt{3}}$:
$$a = 4\sqrt{3}\times \frac{2}{\sqrt{3}} = 8 \text{ cm}$$
Final answer: 8 cm.
Example 5
An equilateral triangle has an area of $9\sqrt{3}$ cm². Find its height.
Use the area formula to find the side:
$$A = \frac{\sqrt{3}}{4}a^2$$
$$9\sqrt{3} = \frac{\sqrt{3}}{4}a^2$$
Divide both sides by $\sqrt{3}$, then multiply by 4:
$$9 = \frac{a^2}{4} \quad\Rightarrow\quad a^2 = 36 \quad\Rightarrow\quad a = 6 \text{ cm}$$
Apply the height formula:
$$h = \frac{\sqrt{3}}{2}\times 6 = 3\sqrt{3} \approx 5.20 \text{ cm}$$
Final answer: $3\sqrt{3} \approx 5.20$ cm.
Example 6
A triangular road sign is equilateral with each side 0.5 m. How tall is the sign?
$$h = \frac{\sqrt{3}}{2}\times 0.5$$
$$h = 0.25\sqrt{3} \approx 0.433 \text{ m}$$
Final answer: about 0.433 m, roughly 43 cm tall.
Why the Height of an Equilateral Triangle Matters
"Drop the altitude and you get a 30-60-90 triangle."
That is the real WHY. The altitude doesn't just measure the triangle's height — it manufactures the most important right triangle in all of geometry: the 30-60-90 triangle, whose side ratio $1 : \sqrt{3} : 2$ is the source of $\sin 60° = \sqrt{3}/2$ and $\cos 30° = \sqrt{3}/2$. The equilateral triangle is where those exact trig values are born.
Where the height does real work:
Trigonometry's special angles. Splitting an equilateral triangle is the standard way to derive the sine and cosine of 30° and 60° — no calculator, no memorising. Once you can draw the height, you can rebuild those values from scratch.
Trusses, signage, and packing. Equilateral triangles tile the plane and pack efficiently; their height sets the row spacing in triangular and hexagonal lattices, from honeycomb to steel truss panels.
Area from a single number. Because $A = \tfrac{\sqrt{3}}{4}a^2$ and $h = \tfrac{\sqrt{3}}{2}a$ both depend only on the side, one measurement fixes the entire triangle — a property engineers lean on for quick equilateral-component sizing.
These properties tie back to the broader properties of a triangle, specialised to the all-equal case.
Where Students Trip Up on the Height of an Equilateral Triangle
Mistake 1: Treating the height as half the side
Where it slips in: Finding the height straight from the side length.
Don't do this: Set $h = a/2$ because "the altitude splits the triangle in half."
The correct way: The altitude bisects the base (that gives the $a/2$ leg). The height is the perpendicular leg, found by Pythagoras: $h = \tfrac{\sqrt{3}}{2}a$, about 0.866 of the side.
The first-instinct error is confusing "the altitude cuts the base in half" with "the height is half the side." Drawing the right triangle once — hypotenuse $a$, base leg $a/2$, vertical leg $h$ — makes it clear that $h$ is the unknown leg, not a halved side.
Mistake 2: Forgetting to halve the base before applying Pythagoras
Where it slips in: Deriving or re-deriving the height from scratch.
Don't do this: Use the full base $a$ as a leg of the right triangle, writing $a^2 = h^2 + a^2$.
The correct way: The altitude meets the base at its midpoint, so the leg is $a/2$, giving $a^2 = h^2 + (a/2)^2$.
The rusher who skips drawing the altitude is the one who uses the full base — and gets the impossible $h = 0$. The half-base step is the one that looks skippable but isn't; it is exactly where the $\sqrt{3}$ comes from.
Mistake 3: Mixing up the height formula with the area formula
Where it slips in: Recalling the formula under time pressure.
Don't do this: Write $h = \tfrac{\sqrt{3}}{4}a$ (borrowing the area's coefficient) or square the side in the height formula.
The correct way: Height is linear in the side: $h = \tfrac{\sqrt{3}}{2}a$. Area is quadratic: $A = \tfrac{\sqrt{3}}{4}a^2$. The denominators (2 vs 4) and the power of $a$ (1 vs 2) are the tells.
Key Takeaways
The height of an equilateral triangle with side $a$ is $h = \dfrac{\sqrt{3}}{2}a$ — about 0.866 of the side.
The formula is derived by applying Pythagoras to the right triangle the altitude forms, using a leg of $a/2$.
From the perimeter, $a = P/3$; from the area, solve $A = \tfrac{\sqrt{3}}{4}a^2$ for $a$ first, then find $h$.
The most common error is treating the height as half the side — it is $\sqrt{3}/2$ of the side, not $1/2$.
Splitting an equilateral triangle gives the 30-60-90 triangle, the origin of $\sin 60° = \sqrt{3}/2$.
A Practical Next Step
Practice these problems to solidify your understanding.
Question 1: Find the height for side 14 cm.
Question 2: A perimeter is 30 cm — find the height.
Question 3: The height is $7\sqrt{3}$ cm — find the side. If you get stuck on Question 2, return to "Finding the Height from the Area or Perimeter" and convert the perimeter to a side first.
Want a live Bhanzu trainer to walk through more equilateral-triangle problems? Book a free demo class.
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