How to Find the Height of an Equilateral Triangle

Geometry Lessons

The short answer

h = (√3 / 2) x s

Multiply the side length by about 0.866. For a side of 10 the height is 8.66. The halving of the base is what makes this work, and it is the step people skip.

An equilateral triangle has no obvious height, because no side sits flat against another and nothing is drawn square to anything. Drop a perpendicular from any corner though, and it lands exactly on the midpoint of the opposite side. That single fact is what everything else here depends on.

Why the altitude lands on the midpoint

All three sides are equal and all three angles are 60 degrees. Drop the perpendicular from the apex and the two halves you create are mirror images of each other: they share the altitude, their hypotenuses are both full sides of the original, and both have a right angle at the foot. That makes them congruent, so the two base segments have to be equal.

It is worth being precise about this, because the fact is often remembered too loosely. The base halves here only because the two sides meeting at the apex are equal. On a scalene triangle the perpendicular lands wherever it lands, and assuming a midpoint will quietly give you the wrong answer.

1

Halve the base

The altitude meets it at the midpoint, so the right triangle you build uses half the side, not the whole one.

2

Apply Pythagoras

The hypotenuse is the full side. The height squared is the side squared minus half the side squared.

3

Simplify

What is left is three quarters of the side squared, and its square root is root three over two times the side.

A worked example

Take a triangle with every side 10 units. Half the base is 5. Pythagoras gives the height squared as 100 minus 25, which is 75, so the height is the square root of 75, or 8.66 units.

Check it against the area. Half of 10 times 8.66 is 43.30, and the standard equilateral area formula, root three over four times the side squared, also gives 43.30. Two routes, same number.

An equilateral triangle with every side 10 units, its altitude drawn as a dashed line splitting the base into two segments of 5, beside the four steps taking Pythagoras from a hypotenuse of 10 and a base of 5 to a height of 8.66.
The whole method sits in that split base. Once you accept the altitude cuts the base into 5 and 5 rather than leaving it at 10, the rest is a single application of Pythagoras.

The base halves, the side does not

Using the full side as the base of the right triangle is the error that catches people, and it gives itself away immediately: you end up subtracting a number from itself and the height comes out as zero. If you ever see zero here, this is why.

base 10givesh² = 10² – 10²0

the full side was used as the base of the right triangle

base 5givesh² = 10² – 5²8.66

the altitude bisects the base, so the right triangle uses 5

Where the root three comes from

It is not arbitrary. Subtracting a quarter of the squared side from the whole squared side always leaves three quarters of it, whatever the side length happens to be. The square root of three quarters is root three over two, so the ratio between an equilateral triangle and its height is fixed at about 0.866 no matter the size. That is why you can multiply straight through instead of redoing Pythagoras each time.

Going backwards from the height

If you know the height and want the side, rearrange to s = 2h divided by root three. A height of 8.66 gives 2 times 8.66 divided by 1.732, which is 10. This comes up whenever a drawing gives you a vertical dimension and leaves the side lengths off.

Mistakes that catch people out

  • Using the full side as the base of the right triangle. Halve it first. Not halving gives a height of zero.
  • Confusing the height with the side. The height is always shorter, at about 87 percent of the side.
  • Assuming any altitude bisects the base. It only does so when the two sides meeting at the apex are equal.
  • Rounding root three too early. Use 1.732 at minimum, and prefer the full calculator value.
  • Mixing up the height and area formulas. Height uses root three over two. Area uses root three over four.

Practice questions

  1. A side of 6. Find the height.
  2. A side of 12. Find the height.
  3. A side of 4. Find the height.
  4. A height of 12.99. Find the side.
  5. A side of 20. Find the area.
Show the answers
  1. h = 6 x 0.866 = 5.196 units
  2. h = 12 x 0.866 = 10.392 units
  3. h = 4 x 0.866 = 3.464 units
  4. s = 2 x 12.99 / 1.732 = 15 units
  5. h = 17.32, then ½ x 20 x 17.32 = 173.2 square units

Common questions

What is the height of an equilateral triangle?

It is the side length times the square root of three, divided by two. In decimals, multiply the side by about 0.866.

Why is the base halved?

Because the altitude of an equilateral triangle lands exactly on the midpoint of the opposite side. The two halves it creates are congruent right triangles, so the base splits evenly.

Where does the square root of three come from?

From simplifying Pythagoras. Subtracting a quarter of the squared side from the whole squared side leaves three quarters, and the square root of three quarters is root three over two.

Is the height the same as the side?

No, and it is always shorter. The height is about 86.6 percent of the side length, whatever the size of the triangle.

How do I find the side if I know the height?

Rearrange to s = 2h divided by the square root of three. A height of 8.66 gives a side of 10.

What is the difference between the height and area formulas?

The height uses root three over two times the side. The area uses root three over four times the side squared. The two are easy to confuse.

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