Equilateral Triangle Calculator

What makes a triangle equilateral

An equilateral triangle has three sides of the same length and three angles of the same size. Each angle is 60 degrees, and 60 + 60 + 60 = 180, which is the total every triangle has to reach. Because that one rule locks the shape down, a single number describes it. Give the calculator the side, the height, the area, the perimeter or either circle radius, and the other five follow.

The symmetry is stronger than it looks. The altitude from any corner lands exactly on the middle of the opposite side, so it is also a median, and it cuts that corner angle in half, so it is also an angle bisector. All three of those lines are the same line. That is why one split gives you everything: a right triangle with legs of s/2 and h and a hypotenuse of s holds the height, the area and both circle radii.

Six of these triangles around a shared center make a regular hexagon.

The height: split the triangle in half

Drop a line from the top corner straight down to the base. The base splits into two equal parts of s/2 and two right triangles appear. Pythagoras on either one says (s/2) squared plus h squared equals s squared. Move the first term across and h squared = s squared - s squared/4 = 3s squared/4, so h = s x the square root of 3 divided by 2, which is 0.8660254 x s.

With a side of 12 ft: 12 x 0.8660254 = 10.3923 ft. The reverse check works too. Half the base is 6 ft, and 6 squared plus 10.3923 squared = 36 + 108 = 144, which is 12 squared.

Two things follow. The height is always 86.6025 percent of the side, so a 12 ft triangle fits under a 10.5 ft ceiling with room to spare. And the height is never half the side: half of 12 is 6, while the real height is 10.3923.

Area: 0.4330127 times the side squared

Area is half the base times the height. The base is s and the height is s x 0.8660254, so the area is s squared x 0.4330127. Both the height and the area use the side alone, which is why this page never asks for an angle.

For the 12 ft triangle: 144 x 0.4330127 = 62.3538 square feet, and 3 x 12 = 36 ft of perimeter. The square that would sit around it, 12 ft by 12 ft, holds 144 square feet, so the triangle fills about 43 percent of it. Scale the side and the area moves much faster: double 6 ft to 12 ft and the area goes from 15.5885 to 62.3538 sq ft, four times bigger, because 2 squared = 4.

Working backward from area, height, perimeter or a radius

Most real jobs hand you something other than the side. A lot listing gives the area, a cut gives the height, a roll of trim gives the perimeter, and each formula turns around with one algebra step.

From the area: s = the square root of (4 x A / 1.7320508). With A = 50 sq ft, 4 x 50 = 200 and 200 / 1.7320508 = 115.4701, whose square root is 10.7457 ft.

From the height: s = 2h / 1.7320508. At h = 9 in, 18 / 1.7320508 = 10.3923 in. From the perimeter: s = P / 3, so 45 cm of trim means 15 cm sides, a 12.9904 cm height and 97.4279 sq cm of area.

From the circles: s = R x 1.7320508, so a triangle drawn inside a 10 in circle has 17.3205 in sides and a height of exactly 15 in. The inradius version is s = 2 x r x 1.7320508, so a 4 in inradius gives 13.8564 in sides and a 12 in height.

The two circles that fit the triangle

Every triangle has one circle inside it that touches all three sides, the incircle, and one circle through all three corners, the circumcircle. For an equilateral triangle both are centered on the same point, the spot where the three altitudes meet, which is also the balance point of the shape.

The inradius is s x 0.2886751. The circumradius is s x 0.5773503. Divide one by the other and the side cancels: R = 2r, every time, at every size. With a 12 ft side the incircle has a 3.4641 ft radius and the circumcircle has 6.9282 ft, and 2 x 3.4641 = 6.9282.

The inradius also carries the name apothem, the distance from the center to the middle of a side. That number does extra work in a regular hexagon, which is six of these triangles sharing a center: the hexagon's apothem is the triangle's inradius and its side is the triangle's side.

A 12 ft triangular bed, start to finish

Say you are marking out a corner bed with three 12 ft sides. Stretch a string for one side, measure 12 ft along it, then swing two more 12 ft lengths from each end until they meet. The corner where they meet is 60 degrees, and the perpendicular from that corner to the opposite side should measure 10.3923 ft.

Edging runs the perimeter, 36 ft. Soil at 6 in deep needs 62.3538 sq ft x 0.5 ft = 31.1769 cubic feet, and bags sold at 1.5 cu ft mean 31.1769 / 1.5 = 20.7846, so buy 21 bags. Fence posts every 4 ft around a closed loop: 36 / 4 = 9 gaps, and a closed loop needs one post per gap, so 9 posts rather than 10. Plants on a 2 ft grid take 4 sq ft each: 62.3538 / 4 = 15.5884, so 15 plants fit with slack at the corners.

Move the side to 10 ft and the calculator redoes all of it: 43.3013 sq ft of area, an 8.6603 ft height, 30 ft of edging and 30 / 4 = 7.5, which rounds up to 8 posts on the smaller loop.

Where 0.8660254 and 0.4330127 come from

Both constants are the square root of 3 in disguise. The square root of 3 is 1.7320508, so half of it is 0.8660254 and a quarter of it is 0.4330127. The decimal is fine for building and gardening.

Rounding early is the usual source of a wrong answer. Round the area constant to 0.433 and a 12 ft triangle comes out at 144 x 0.433 = 62.352 sq ft, which is 0.0018 sq ft short of 62.3538. That gap does not matter for soil. Round harder, to 0.43, and the same triangle gives 144 x 0.43 = 61.92, off by 0.4338 sq ft, close to half a bag over a large bed. Round the height to 0.87 and 12 x 0.87 = 10.44, which runs 0.0477 ft long, about half an inch.

Units follow the usual rules: a length squared gives a square unit, so feet become square feet. NIST publishes the Guide for the Use of the International System of Units (NIST Special Publication 811, 2008 edition) covering how those units are written and abbreviated. Keep the unit on the input and on the output, and do not mix inches with feet on the same line.

Equilateral compared with other triangles

A right triangle with legs of 10.3923 ft and 6 ft covers 0.5 x 10.3923 x 6 = 31.1769 sq ft, exactly half of the 12 ft equilateral triangle, because it is one of the two halves cut by the altitude. An isosceles triangle with the same 12 ft base but a 9 ft height covers 0.5 x 12 x 9 = 54 sq ft. Same footprint, less area.

If the three sides are not all equal, or you are mixing sides with angles, this page is the wrong tool. Use the general triangle calculator, which runs the law of cosines and the law of sines. If the triangle has a 90 degree corner, the right triangle calculator handles the legs, the hypotenuse and the trig ratios.

One sanity rule holds for any answer here: the height should land near 0.8660254 x s, and the area a bit under half of s squared.

The Monash University maths support notes on perimeter and area (Monash University, checked October 2026) walk the same 12 cm case, getting a 36 cm perimeter and 62.35 sq cm of area, and the Math 1330 section 7.2 notes (University of Houston, checked October 2026) reach the same area formula from half of s squared times the sine of 60 degrees.