Height of Equilateral Triangle: Formula, Proof, Examples

TL;DR

The height of an equilateral triangle with side a is ( h = \frac{\sqrt{3}}{2} a ). It comes from the Pythagorean theorem applied to the right triangle the altitude creates. This article gives the formula, its derivation, methods for finding height from the side, area, or perimeter, and six worked examples.

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 derived by applying 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 ( \frac{a}{2} ) (half the base, because the altitude bisects it).

Applying the Pythagoras theorem:

[ a^2 = h^2 + \left( \frac{a}{2} \right)^2 ]

Isolate h:

[ h^2 = a^2 - \frac{a^2}{4} = \frac{4a^2 - a^2}{4} = \frac{3a^2}{4} ]

Take the positive square root:

[ h = \frac{\sqrt{3}}{2} a ]

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 ), we can find ( a = \frac{P}{3} ). Substitute into the height formula: ( h = \frac{\sqrt{3}}{2} \cdot \frac{P}{3} = \frac{\sqrt{3} P}{6} ).
• From the area (A): The area of an equilateral triangle is ( A = \frac{\sqrt{3}}{4} a^2 ). Solve for a first, then apply the height formula — or use the direct relation ( h = \frac{2A}{a} ) once the side is known.

In every case, find the side a first, then run the height formula.

Examples of Height of Equilateral Triangle

Example 1

Find the height of an equilateral triangle with side 6 cm.

[ h = \frac{\sqrt{3}}{2} \times 6 \approx 5.20 \text{ cm} ]

Example 2

Find the height of an equilateral triangle with side 10 cm.

[ h = \frac{\sqrt{3}}{2} \times 10 \approx 8.66 \text{ cm} ]

Example 3

An equilateral triangle has a perimeter of 18 cm. Find its height.

Find the side from the perimeter: ( a = \frac{P}{3} = \frac{18}{3} = 6 ext{ cm} )

Then apply the height formula:

[ h = \frac{\sqrt{3}}{2} \times 6 \approx 5.20 ext{ 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:

[ 4\sqrt{3} = \frac{\sqrt{3}}{2} a \Rightarrow a = 8 ext{ cm} ]

Example 5

An equilateral triangle has an area of ( 9\sqrt{3} ) cm². Find its height.

Use the area formula to find a:

[ 9 = \frac{\sqrt{3}}{4} a^2 \Rightarrow a = 6 ext{ cm} ]

Then apply the height formula:

[ h = \frac{\sqrt{3}}{2} \times 6 \approx 5.20 ext{ 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 \approx 0.433 ext{ m} ]

Why the Height of an Equilateral Triangle Matters

The altitude doesn't just measure the triangle's height — it creates the important right triangle in geometry: the 30-60-90 triangle. The equilateral triangle is where those exact trig values are born.

Where Students Trip Up on the Height of an Equilateral Triangle

Mistake 1: Treating the height as half the side

The altitude bisects the base but the height itself is found by Pythagoras: ( h = \frac{\sqrt{3}}{2} a ).

Mistake 2: Forgetting to halve the base before applying Pythagoras

Using the full base a results in incorrect calculations. Correctly use ( \frac{a}{2} ).

Mistake 3: Mixing up the height formula with the area formula

Height is linear in the side, ( h = \frac{\sqrt{3}}{2} a ), while area is quadratic, ( A = \frac{\sqrt{3}}{4} a^2 ).

Key Takeaways

• The height of an equilateral triangle is ( h = \frac{\sqrt{3}}{2} a ) — about 0.866 of the side.
• The formula is derived using Pythagorean theorem from the right triangle created by the altitude.
• Common errors include treating height as half the side length.

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 ( 73\sqrt{3} ) cm — find the side.