Equilateral triangle measures
Find the height and area of an equilateral triangle from one side.
Equilateral triangle measures is one of 5 triangles formulas in the geometry section of this library, and it is used at middle school · high school level.
Why equilateral triangle measures works
Cutting an equilateral triangle straight down the middle produces two right triangles whose hypotenuse is a full side and whose short leg is half a side. The Pythagorean theorem then gives the height, and feeding that height into the half-base-times-height rule gives the area.
What each symbol means
$s$ is the common side length.
Equilateral triangle measures: when it holds
$s\ge0$ and all three sides must be equal.
When it stops applying
All three sides have to be equal. In an isosceles triangle with only two equal sides, the height depends on the length of the odd side, and no fixed multiple of a side gives it.
Equilateral triangle measures: a worked example
For $s=6$, $h=3\sqrt3$ and $A=9\sqrt3$.
The mistake to avoid
What people do: Using a side length as though it were the height.
Why it goes wrong: The height is only about 0.866 of a side, so for a side of 6 that shortcut reports 18 square units when the true area is about 15.59.
Do this instead: Multiply the side by the square root of 3 and halve it to get the height first, then apply the ordinary triangle rule.
Equilateral triangle measures: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $s$ is the common side length.
- Check the conditions before substituting. $s\ge0$ and all three sides must be equal.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Triangles
- Level
- Middle school · High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where equilateral triangle measures comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Area of a Triangle — the lesson behind this formula: choose a base-height, trigonometric, or side-length formula.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about equilateral triangle measures
Where does the square root of 3 come from?
From the half triangle: a full side squared minus half a side squared leaves three quarters of the side squared, and taking the root introduces it.
What are the angles involved?
All three corners are 60 degrees, and the half triangle created by the height is the familiar 30-60-90 shape.
How do I recover the side from the area?
Multiply the area by 4, divide by the square root of 3, and take the square root. An area of 9 times the root of 3 returns a side of 6.
Is the height the same as the distance from centre to side?
No. The centre sits one third of the way up, so that distance is a third of the height, which is the side times the root of 3 divided by 6.