What is the height of an equilateral triangle?

The side times the square root of 3, divided by 2, or about 0.866025 times the side. A side of 6 has a height of 5.19615, which is 3 times the square root of 3. It follows from Pythagoras: the height cuts the triangle into two right triangles with a hypotenuse of 6 and a base of 3, so the height squared is 36 minus 9, which is 27. All three heights are the same length, and the tool lists each one.

Why are two angles already filled in?

Because the tool solves a triangle from three values, and one side with two 60 degree angles is three values that describe exactly one triangle: an equilateral one. Change only the side and it stays equilateral at the new size. Change an angle and it is a different triangle, still solved, and the result line names its type, isosceles or scalene, so nothing is quietly kept equilateral that is not.

It is worked out in full double precision and shown to six significant figures, so a side of 6 shows 15.5885, which is 9 times the square root of 3. The Copy button gives you the same rounded figures; the side you typed is used as typed, in whatever unit you measured it, and the area is in that unit squared.

Equilateral Triangle Calculator

Type one side of an equilateral triangle to get its height, area and perimeter, with two angles preset to 60 degrees and the figure drawn to scale. The page opens with side a = 6 and angles A and B both at 60, so the side is the only box to change. The other two sides come back as 6, the third angle as 60, the height to each side as 5.19615, the area as 15.5885 and the perimeter as 18, and the last line of the result reads equilateral. The height is the side times the square root of 3 over 2, and the area is the square root of 3 over 4 times the side squared. This page starts from a side length; it does not work backwards from a height, an area or a perimeter.

Solvedone triangle

AASa 6 b 6 c 6Degrees

a 6   b 6   c 6
A 60°   B 60°   C 60°
area 15.5885     perimeter 18
height to a 5.19615   to b 5.19615   to c 5.19615
acute, equilateral

the 180 degree angle sum, then the law of sines for the two missing sides

A (°)

B (°)

C (°)

Angles in

60°

Area

15.5885

Two angles and a side that is not between them, so A, B and a. The third angle is what is left of 180 degrees, and the law of sines scales the triangle to the side you gave.

AAS: one triangle fits.

SSS 7, 8, 9three sides
one triangle, area 26.8328
SSA ambiguousa 7, b 9, A 40°
two triangles, both shown: B 55.7349° or 124.2651°
Impossible sides1, 2, 9
refused: These three lengths cannot meet: 1 and 2 together are shorter than 9

How the five cases work. Fill any three of the six boxes and the tool reads which case you gave it. Three sides go to the law of cosines for every angle and Heron’s formula for the area. Two sides with the angle between them get the third side from the law of cosines first. Two angles and any side get the third angle from the 180 degree sum and the missing sides from the law of sines. Two sides with an angle that is not between them are the ambiguous case: when the side opposite the given angle is shorter than the other side but longer than the height, two different triangles fit, and you get both. Three angles are refused, because they set the shape and not the size. Exact trigonometry by the law of cosines and the law of sines, with rounding clamped so a right angle does not come back as 89.999 degrees. The ambiguous side-side-angle case returns both valid triangles rather than picking one.

Common questions

What is the height of an equilateral triangle?: The side times the square root of 3, divided by 2, or about 0.866025 times the side. A side of 6 has a height of 5.19615, which is 3 times the square root of 3. It follows from Pythagoras: the height cuts the triangle into two right triangles with a hypotenuse of 6 and a base of 3, so the height squared is 36 minus 9, which is 27. All three heights are the same length, and the tool lists each one.
Why are two angles already filled in?: Because the tool solves a triangle from three values, and one side with two 60 degree angles is three values that describe exactly one triangle: an equilateral one. Change only the side and it stays equilateral at the new size. Change an angle and it is a different triangle, still solved, and the result line names its type, isosceles or scalene, so nothing is quietly kept equilateral that is not.
Is the area exact?: It is worked out in full double precision and shown to six significant figures, so a side of 6 shows 15.5885, which is 9 times the square root of 3. The Copy button gives you the same rounded figures; the side you typed is used as typed, in whatever unit you measured it, and the area is in that unit squared.

Exact trigonometry by the law of cosines and the law of sines, with rounding clamped so a right angle does not come back as 89.999 degrees. The ambiguous side-side-angle case returns both valid triangles rather than picking one.