Equilateral Triangle Calculator

A = (√3 ÷ 4) × s² P = 3s h = (√3 ÷ 2) × s

Side

The length of any one of the three sides, in centimetres. This is the only input on the page, because an equilateral triangle has all three sides equal and all three angles fixed at 60 degrees, which leaves the size as the single thing to decide

Area

The space the triangle covers, in square centimetres, which is the square root of three divided by four, times the side squared. The constant is about 0.433, which is a useful number to carry around: an equilateral triangle covers a little over four tenths of the square drawn on its side

Perimeter

The distance all the way around, in centimetres: three times the side, since all three are the same. It is the one output that is always exact whenever the side is, which is what makes it the figure to check the other two against

Height

The distance from a corner straight down to the middle of the opposite side, in centimetres, which is the square root of three over two times the side. It is also called the altitude, and it is the same line whichever corner you start from because the shape is symmetric

√3

The square root of three, about 1.7320508. It appears in both the area and the height and it is not a rounding artefact: splitting the triangle in half leaves a right-angled triangle whose sides cannot all be whole numbers at once, so the odd number has to come out somewhere

Four decimal places

How wide every output is written. The area and the height almost always carry the square root of three and so almost never end; the perimeter is exact, and printing it to four places is a fixed width rather than a claim about precision

The page is for anything made of equilateral triangles, and there are more of those than the name suggests. Roof trusses, bicycle frames, bridge lattice and shelving brackets all use them, because a triangle is the only polygon that cannot be pushed out of shape without changing the length of a side — give a square frame a sideways shove and it folds flat, which is why the diagonal brace on a gate is there. If you are cutting triangles out of a sheet, the area is what tells you how many fit, and the height is what tells you how tall a row of them is: laid point-up in rows, two rows overlap by exactly one height, so the height is the spacing rather than the area. If you are folding a strip into a shape, the perimeter is the length of strip you need before you allow for the join. And if you are checking a drawing, the height has a fixed place in the triangle — it cuts the shape into two identical right-angled triangles and it meets the opposite side exactly at its middle — so a calculated height that does not leave two equal halves means the triangle in the drawing is not equilateral, whatever the label says. One arithmetic warning belongs here rather than in the small print: the height printed on this page has already been rounded to four places, so multiplying it back in by hand gives an area one digit off in the last place. The two figures on the panel are each correct; it is the round trip through the printed height that loses the digit.

1. A side of 6

   1. Perimeter: 3 × 6 = 18
   2. Height: (√3 ÷ 2) × 6 = 5.196152…, which rounds to 5.1962
   3. Area: (√3 ÷ 4) × 36 = 15.588457…, which rounds to 15.5885

   The input the page loads with, and the row to hold on to, because it is the one the other triangle pages meet: a triangle with all three sides 6 has an area of 15.5885 on this page, and the same 15.5885 comes out of the isosceles page at base 6 and leg 6 and out of the Heron page at 6, 6, 6. Three different routes to the same shape, and the three agree to the last digit. The arithmetic trap is also visible here: 6 × 5.1962 ÷ 2 is 15.5886, one digit above the area above, and the digit is lost in the rounding of the height rather than in either formula.

2. A side of 1

   1. Perimeter: 3 × 1 = 3
   2. Height: (√3 ÷ 2) × 1 = 0.8660254…, which rounds to 0.866
   3. Area: (√3 ÷ 4) × 1 = 0.4330127…, which rounds to 0.433

   The unit triangle, and the one to remember the two constants from: an equilateral triangle of side one has a height of 0.8660 and an area of 0.4330. Against the square of side one, whose area is 1, the triangle covers a little under half as much ground for the same edge, which is the plainest way to see where the square root of three over four comes from. Two of the three figures print with trailing zeros dropped, since the fourth decimal is a zero in both.

3. A side of 2

   1. Perimeter: 3 × 2 = 6
   2. Height: (√3 ÷ 2) × 2 = 1.7320508…, which rounds to 1.7321
   3. Area: (√3 ÷ 4) × 4 = 1.7320508…, which rounds to 1.7321

   The row that looks like a mistake and is not: the area and the height both print 1.7321. Both are the square root of three for this side — halving the side cancels the division by two in the height, and squaring it cancels the division by four in the area — but one is 1.7321 square centimetres and the other is 1.7321 centimetres. They are different quantities that happen to share a numerical value at this one side length, and the same thing happens at no other side on the table.

4. A side of 10

   1. Perimeter: 3 × 10 = 30
   2. Height: (√3 ÷ 2) × 10 = 8.660254…, which rounds to 8.6603
   3. Area: (√3 ÷ 4) × 100 = 43.301270…, which rounds to 43.3013

   A ten-centimetre triangle, which is about the size of a roofing bracket. The height of 8.6603 is 86.6 per cent of the side, and that ratio never changes: the height of an equilateral triangle is always 0.866 of its side, whatever the size. If you are laying these triangles out in rows point-up, that is the number to space the rows by, since each row overlaps the one below by exactly one height.

5. A side of 0

   1. Perimeter: 3 × 0 = 0
   2. Height: (√3 ÷ 2) × 0 = 0
   3. Area: (√3 ÷ 4) × 0 = 0

   Zero is a legal input rather than an empty box. A triangle with a side of zero has collapsed to a single point and all three measurements are zero, so a row of zeros is a real answer. Leaving the box blank is the different case: with nothing in it the page shows no result at all, because there is nothing to work from.

The page is for equilateral triangles only: all three sides the same and all three angles 60 degrees. A triangle that is merely isosceles, or right-angled, or any other shape with one side of this length, cannot be described by that one number and the page will give you the equilateral figures for it rather than telling you the shape you have in mind is a different one. If you know two sides of a general triangle, the triangle area page takes a base and a height and the Heron page takes all three sides. The outputs are in centimetres and square centimetres whatever the unit dropdown is set to, so a side entered in inches comes back as a centimetre answer. Four decimal places is a display width rather than a claim about precision, and it is generous for a measured side. The height and the area are printed independently: the area comes from the unrounded height, so multiplying the printed height back in by hand lands one digit out in the fourth decimal, which is a rounding difference rather than an error in either figure. The page is two-dimensional throughout and has no field for the thickness of a physical triangle, so it cannot tell you the volume of a triangular prism or the weight of a triangular plate.

Why is one measurement enough to describe an equilateral triangle?

Because the shape is fixed before you start. An equilateral triangle has all three sides the same length and all three angles at 60 degrees, so the only thing left to choose is how big it is, and the side length settles that. A general triangle has no such freedom used up: one side leaves infinitely many shapes, and you need three sides, or two sides and the angle between them, before the triangle is pinned down.

How is the height worked out, and where does the square root of three come from?

Dropping the height from a corner cuts the triangle into two identical right-angled triangles. Each has a hypotenuse equal to the side and a base of half the side, so Pythagoras gives the height as the square root of the side squared minus a quarter of the side squared, which is the square root of three over two times the side. The odd number is not a rounding artefact; it is what is left when a shape with equal sides is cut in half.

Multiplying the printed height by half the base does not give exactly the printed area. Why?

Because the height on the panel has already been rounded to four decimal places, and the area was computed from the unrounded height and then rounded on its own. At a side of 6 the height prints as 5.1962, half the base is 3, and the product is 15.5886, while the area prints as 15.5885. Both figures are right; the fourth digit goes missing in the round trip, not in either calculation. If you need the exact area, use the area row rather than rebuilding it.

What is the height as a fraction of the side?

It is always 0.866 of the side, about 86.6 per cent, whatever the size of the triangle. That follows from the same square root of three over two: increasing the side scales the height in step, so the ratio between them never changes. It is the number to use when laying triangles out in rows, since each row of point-up triangles overlaps the one below by exactly one height.

How does the area compare with a square of the same side?

The square is bigger, at 1 square unit against about 0.433 for the triangle at a side of 1. The triangle uses less material for the same edge, but it also has a shorter perimeter over that area, so the comparison only means something stated as material per unit of edge. In practice the reason to pick a triangle is stiffness rather than area: a triangle cannot be deformed without changing the length of a side, which a square can.

Can the side be zero, and is that the same as leaving the box empty?

Zero is a real input and all three outputs come back as zero, which is the honest answer for a triangle with no size. A blank box is different: with nothing entered the page shows no result at all, because there is nothing to work from. That distinction matters while you are typing, since clearing the box to retype a number should not flash up a row of zeros.

• Equilateral triangle — the three-sided regular polygon, with the height, the area and the 60-degree angles this page computes, and the split into two right-angled triangles that produces the square root of three — Wolfram MathWorld (United States)
• Triangle — the general case this page is the simplest member of, with the three-sides and two-sides-and-an-angle conditions that show why a general triangle needs more than one number to pin down — Wolfram MathWorld (United States)
• Regular polygon — the family an equilateral triangle belongs to as the three-sided member, where the side count sets the angles: 60 degrees here, 108 in a pentagon and 135 in an octagon — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); the sides, perimeter and area of a triangle are part of the compulsory-education mathematics curriculum, and the Pythagorean theorem falls in the same band, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部