About

An equilateral triangle is the only regular polygon with three sides. All internal angles equal 60°. A single measurement - the side length a - determines every other property: area, height, perimeter, inradius, and circumradius. Errors in computing triangle areas propagate directly into material estimates, structural load calculations, and CNC cutting paths. This calculator uses the closed-form formula A = √34 ⋅ a² and supports reverse computation from any known property back to the side length. Results assume Euclidean geometry on a flat plane. The tool does not account for surface curvature at geodetic scales.

Formulas

All properties of an equilateral triangle derive from its side length a. Below are the exact closed-form expressions used in this calculator.

A = √34 ⋅ a²

where A is the area and a is the side length.

h = √32 ⋅ a

where h is the altitude (height) from any vertex to the opposite side. In an equilateral triangle, the altitude, median, angle bisector, and perpendicular bisector all coincide.

P = 3 ⋅ a

where P is the perimeter.

r = a ⋅ √36

where r is the inradius (radius of the inscribed circle).

R = a ⋅ √33

where R is the circumradius (radius of the circumscribed circle). Note that for an equilateral triangle, R = 2r always holds.

Reverse formulas used when computing side from a known property:

a = √4 ⋅ A√3

(side from area)

a = 2 ⋅ h√3

(side from height)

Reference Data

Side Length (a)	Area (A)	Height (h)	Perimeter (P)	Inradius (r)	Circumradius (R)
1	0.4330	0.8660	3	0.2887	0.5774
2	1.7321	1.7321	6	0.5774	1.1547
3	3.8971	2.5981	9	0.8660	1.7321
4	6.9282	3.4641	12	1.1547	2.3094
5	10.8253	4.3301	15	1.4434	2.8868
6	15.5885	5.1962	18	1.7321	3.4641
7	21.2176	6.0622	21	2.0207	4.0415
8	27.7128	6.9282	24	2.3094	4.6188
9	35.0740	7.7942	27	2.5981	5.1962
10	43.3013	8.6603	30	2.8868	5.7735
12	62.3538	10.3923	36	3.4641	6.9282
15	97.4279	12.9904	45	4.3301	8.6603
20	173.2051	17.3205	60	5.7735	11.5470
25	270.6330	21.6506	75	7.2169	14.4338
30	389.7114	25.9808	90	8.6603	17.3205
40	692.8203	34.6410	120	11.5470	23.0940
50	1082.5318	43.3013	150	14.4338	28.8675
75	2435.6965	64.9519	225	21.6506	43.3013
100	4330.1270	86.6025	300	28.8675	57.7350
200	17320.5081	173.2051	600	57.7350	115.4701

Frequently Asked Questions

The generic formula A = 12 ⋅ b ⋅ h still applies. In an equilateral triangle, the height is h = √32 ⋅ a. Substituting this into 12 ⋅ a ⋅ h yields exactly √34 ⋅ a². The specialized form eliminates one variable, requiring only the side length.

JavaScript uses IEEE 754 double-precision floats, giving approximately 15 - 17 significant decimal digits. For side lengths between 1e-150 and 1e+150, results remain accurate. Beyond that range, you may encounter overflow (Infinity) or underflow (zero). This calculator caps input at 1e+15 and rejects values below 1e-10 to stay well within safe precision bounds. Results are displayed rounded to 4 - 10 decimal places depending on magnitude.

Yes. Select "Area" as the input mode in this calculator. The reverse formula is a = √4A√3. This is derived by algebraically isolating a from the area formula. All other properties (height, perimeter, inradius, circumradius) are then computed from the recovered side length.

For an equilateral triangle, the circumradius is always exactly twice the inradius: R = 2r. This is a unique property not shared by other triangle types. The centroid, incenter, circumcenter, and orthocenter all coincide at the same point, located at 13 of the height from the base.

This calculator works in pure numeric units. If you enter the side in centimeters, the area is in cm², the perimeter in cm, and radii in cm. Switching to inches means all outputs are in in and in². No internal conversion occurs. You must ensure consistent units. A common mistake is mixing meters and centimeters, which produces results off by a factor of 10000 for area.

No. The formula A = √34 ⋅ a² assumes Euclidean (flat) geometry. On a sphere, the sum of interior angles exceeds 180° and the area includes a spherical excess term. On a hyperbolic surface, angles sum to less than 180°. For terrestrial surveying at scales above a few kilometers, geodetic corrections become necessary.