Look a little closer.

One side determines every measurement. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Equilateral triangles tile the plane without gaps.

An altitude bisects the opposite side and creates two 30–60–90 right triangles. The centroid, circumcentre, incentre and orthocentre coincide. The height is √3/2 times the side.

02 / A different perspective

The circumradius is twice the inradius.

An equilateral triangle has three equal sides. In Euclidean geometry this also makes all three interior angles equal to 60 degrees. Specifying one positive side length fixes its size.

03 / A connection to explore

The sum of perpendicular distances from any interior point to the three sides equals the altitude.

Use the forward calculator for triangular tiles and the inverse tools when a design specifies area, total edge length or height. All four tools use the same exact relationships before rounding the results.

Try the idea with numbers

Open the from the side calculator, note its outputs, then multiply each length input by two. Area should increase by four and every reported length should double, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Equilateral triangle at Wolfram MathWorld.