What this calculator does
Most triangle tools go one way: give them a base and a height and they hand back an area. This one runs the other direction. If you already know the area and the base, this calculator finds the height of a triangle by rearranging area = ½ × base × height into height = 2 × area ÷ base.
An equilateral triangle is a special case worth its own path. Because all three sides and angles are equal, its height splits it into two identical right triangles, and height of equilateral triangle reduces to a single formula: height = side × √3 ÷ 2, with no area figure needed at all.
The formula
Choose which two values you know. From area and base, height is simply twice the area divided by the base, undoing the usual area formula. For an equilateral triangle, entering the side length alone is enough, since the ratio between a side and its height is fixed by geometry: the perpendicular height bisects the base and the apex angle, giving a 30-60-90 right triangle where the height is side × √3 ÷ 2.
| Term | Meaning |
|---|---|
| Height | The perpendicular distance from a vertex straight down to the opposite side (or its extension), also called the altitude. |
| Base | The side the height is measured against. Any side can be chosen as the base; the height changes to match. |
| Equilateral triangle | A triangle with all three sides equal and all three angles at 60°, where height = side × √3 ÷ 2. |
The inputs explained
| Field | What to enter |
|---|---|
| Known values | Choose "Area and base" if you know the area and the side you are measuring against, or "Equilateral triangle" if all three sides are equal and you only know one side length. |
| Area (if solving from area and base) | The triangle’s area, only used when solving from area and base. |
| Base (if solving from area and base), or side length (if equilateral) | The base length (area-and-base mode), or the equilateral triangle’s side length (equilateral mode). |
When to use it
Given the area and a base from a survey or plan
Land surveys and building plans often record a triangular area and one side, but not the height itself. Rearranging the area formula recovers it directly, without re-measuring anything.
Working with an equilateral triangle
Equilateral shapes turn up constantly in design, trusses and tiling patterns. Knowing the height from the side alone (or vice versa) avoids treating it as if it were an arbitrary triangle needing three separate measurements.
Checking a height figure someone else supplied
If a height is quoted alongside an area and base, running the same two numbers through this calculator confirms the third figure is consistent, catching a transposed or mistyped measurement.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Height of an equilateral triangle by side length
A range of equilateral triangle side lengths, and the height each one produces.
| Side length | Height | Area (equilateral, from this side) |
|---|---|---|
| 4 | 3.464 | 6.928 |
| 6 | 5.196 | 15.588 |
| 8 | 6.928 | 27.713 |
| 10 | 8.660 | 43.301 |
| 12 | 10.392 | 62.354 |
| 20 | 17.321 | 173.205 |
Height from area and base, at a fixed base length
A fixed base of 10, against a range of known areas.
| Area | Height | Check: ½ × base × height |
|---|---|---|
| 20 | 4.000 | 20.000 |
| 35 | 7.000 | 35.000 |
| 50 | 10.000 | 50.000 |
| 75 | 15.000 | 75.000 |
| 100 | 20.000 | 100.000 |
| 150 | 30.000 | 150.000 |
Questions
How to find the height of a triangle without the area?
If you know two sides and the angle between them, or all three sides, other tools (such as this site’s triangle-area and Heron’s-formula calculators) can find the area first, which this calculator can then turn into a height against any chosen base.
Does it matter which side I call the base?
Yes. A triangle has three different possible base-and-height pairs, one for each side, and they all give the same area but different height values. Be clear about which side the height is measured against before comparing figures.
How to find height of triangle for a right triangle specifically?
For a right triangle, the two legs already act as base and height for each other; the height relative to the hypotenuse is a separate, smaller value equal to (leg₁ × leg₂) ÷ hypotenuse.
Why does the equilateral formula not need the area?
Because every equilateral triangle of a given side length has exactly the same shape, just scaled, so height and side length are always in the same fixed ratio (√3 ÷ 2). Nothing else needs to be known.
For the reverse calculation, area from a base and height you already have, see the triangle area calculator. To classify a triangle as acute, right or obtuse from its three sides, see the triangle calculator.