An equilateral triangle has three equal sides and three 60° angles, so a single measurement fixes everything about it. This calculator starts from whichever number you have (the side, the height, the perimeter, the area, or the radius of the inscribed or circumscribed circle) and returns all the others, with the arithmetic shown.
How to use the equilateral triangle calculator
- Pick what you know from the I know the menu: Side length (s), Height (h), Perimeter (P), Area (A), Inradius (r) or Circumradius (R).
- Type the number in Value. Area is in square units; every other option is a length.
- Choose the Units, from millimeters to miles. The area uses the matching square unit.
- Read the tape. The headline is the area, or the side length when you started from the area. Below it you get the side, height, perimeter, area, both radii and each angle (60°, or 1.047198 rad). Show the work lists every formula with your numbers.
Equilateral triangle formulas
Everything follows from the side s:
When you start from something else, the calculator converts it to a side first:
| You know | Side length |
|---|---|
| Height h | s = 2h ÷ √3 |
| Perimeter P | s = P ÷ 3 |
| Area A | s = √(4A ÷ √3) |
| Inradius r | s = 2√3 · r |
| Circumradius R | s = √3 · R |
Where √3 ÷ 2 and √3 ÷ 4 come from
Drop the altitude from the top corner. It lands on the midpoint of the base and splits the triangle into two 30-60-90 right triangles, each with hypotenuse s and short leg s/2. By the Pythagorean theorem, h² = s² − (s/2)² = ¾s², so h = (√3 ÷ 2)s. The area is half the base times the height: ½ · s · (√3 ÷ 2)s = (√3 ÷ 4)s². In other words, an equilateral triangle covers about 43.3% of the square drawn on the same side.
Worked example: a triangular flower bed
A landscaper wants an equilateral flower bed with 30 ft² of planting area. Choose Area (A), enter 30 and set the units to ft.
- Side: s = √(4 × 30 ÷ 1.732051) ≈ 8.32 ft, the length of each edge board.
- Perimeter: 3 × 8.3236 ≈ 24.97 ft of edging in total.
- Height: 0.866025 × 8.3236 ≈ 7.21 ft from any side to the opposite corner.
- Inradius ≈ 2.40 ft: the largest circle that fits inside is about 4.8 ft across, useful for centering a round paver or fountain.
- Circumradius ≈ 4.81 ft: a sprinkler at the center needs that much reach to wet all three corners.
One center, four names
In most triangles the centroid (where the medians cross), the incenter (center of the inscribed circle), the circumcenter (center of the circle through the corners) and the orthocenter (where the altitudes cross) are four different points. In an equilateral triangle they coincide, because each altitude is also a median, an angle bisector and a perpendicular bisector. The shared center sits one third of the way up each altitude, which gives the tidy relations r = h ÷ 3, R = 2h ÷ 3 and R = 2r.
Common values
| Side s | Height h | Area A | Inradius r | Circumradius R |
|---|---|---|---|---|
| 1 | 0.8660 | 0.4330 | 0.2887 | 0.5774 |
| 2 | 1.7321 | 1.7321 | 0.5774 | 1.1547 |
| 3 | 2.5981 | 3.8971 | 0.8660 | 1.7321 |
| 5 | 4.3301 | 10.8253 | 1.4434 | 2.8868 |
| 10 | 8.6603 | 43.3013 | 2.8868 | 5.7735 |
| 12 | 10.3923 | 62.3538 | 3.4641 | 6.9282 |
| 20 | 17.3205 | 173.2051 | 5.7735 | 11.5470 |
Doubling the side doubles the height and both radii but quadruples the area.
Why builders and nature use it
- Rigidity: a triangle cannot change shape unless a side changes length, and the equilateral form shares loads evenly. Warren trusses in bridges and roof frames are built from equilateral or nearly equilateral panels, and geodesic domes from near-equilateral triangles.
- Tiling: six equilateral triangles meet around a point (6 × 60° = 360°), so they cover a flat surface with no gaps. Six of them also form a regular hexagon, the cell shape of a honeycomb.
- Most area per perimeter: of all triangles with the same perimeter, the equilateral one encloses the most area. A 30 cm perimeter gives about 43.30 cm², while a 7.5–10–12.5 cm right triangle with the same perimeter holds only 37.5 cm².
For two equal sides instead of three, use the isosceles triangle calculator; for squares, hexagons and beyond, the regular polygon calculator; and for a prism with equilateral ends, the triangular prism calculator.
Frequently asked questions
What is the formula for the area of an equilateral triangle?
A = (√3 ÷ 4) × s², where s is the side length. The factor √3 ÷ 4 is about 0.4330, so a triangle with 10 cm sides covers about 43.30 cm².
How do I find the height of an equilateral triangle?
Multiply the side by √3 ÷ 2, which is about 0.8660. A triangle with 6 cm sides is about 5.196 cm tall. Going the other way, the side is 2h ÷ √3, or about 1.1547 times the height.
How do I get the side length from the area?
Use s = √(4A ÷ √3). For 30 ft² of area, s = √(120 ÷ 1.732051) ≈ 8.32 ft. Choose Area (A) in the calculator and the side length becomes the headline result.
Where is the center of an equilateral triangle?
Its centroid, incenter, circumcenter and orthocenter are the same point, one third of the way up each altitude from the base. That is why the inradius is h ÷ 3 and the circumradius is 2h ÷ 3, exactly twice the inradius.
Is every equilateral triangle also isosceles?
Yes. An isosceles triangle needs at least two equal sides, and an equilateral triangle has three. It is also equiangular: every angle is exactly 60°, or π/3 radians.