Equilateral Triangle Calculator

An equilateral triangle is the most symmetric type of triangle, with all three sides equal in length and all three angles measuring exactly 60°. This perfect symmetry makes equilateral triangles particularly elegant mathematically and useful in design, engineering, and nature. From honeycomb patterns to architectural structures, equilateral triangles appear wherever maximum strength and efficiency are needed.

Key Properties

The defining characteristic of an equilateral triangle is that all three sides are equal in length. This immediately implies that all three angles are also equal, and since the sum of angles in any triangle is 180°, each angle must be 180° ÷ 3 = 60°. This makes the equilateral triangle both equilateral (equal sides) and equiangular (equal angles).

Symmetry and Special Properties

Equilateral triangles have three lines of symmetry, each passing through a vertex and the midpoint of the opposite side. These lines of symmetry are also the altitudes, medians, angle bisectors, and perpendicular bisectors - all four of these traditionally different lines coincide in an equilateral triangle. The point where all three lines meet is called the centroid, and it's also the circumcenter, incenter, and orthocenter all at once.

The height of an equilateral triangle is always √3/2 times the side length (approximately 0.866 times the side). This ratio comes from the 30-60-90 triangle formed when you draw the height, which splits the equilateral triangle into two mirror-image 30-60-90 triangles.

Real-World Applications

Structural Engineering: Equilateral triangles are incredibly strong and rigid. Unlike squares or rectangles that can collapse into parallelograms under pressure, triangles maintain their shape. Equilateral triangles distribute stress evenly across all three sides, making them ideal for trusses, bridges, and tower structures like the Eiffel Tower.

Tessellation and Tiling: Equilateral triangles can tile a plane perfectly with no gaps or overlaps. Six equilateral triangles meet at each vertex, and this pattern is found in floor tiles, quilts, and molecular structures. When combined with hexagons (which are made of six equilateral triangles), they create even more complex tessellation patterns.

Nature and Chemistry: Hexagonal patterns in honeycombs are composed of equilateral triangles. Many molecular structures, particularly in chemistry and crystallography, feature equilateral triangle arrangements because they maximize space efficiency while minimizing material use. This is why soap bubbles form hexagonal patterns when packed together.

Common Mistakes to Avoid

• Confusing with isosceles: All equilateral triangles are isosceles (two equal sides), but not all isosceles triangles are equilateral. An equilateral triangle has all three sides equal.
• Forgetting the √3 factor: The height formula includes √3/2, not just 1/2. This is approximately 0.866, not 0.5.
• Thinking angles can vary: In an equilateral triangle, all angles are always exactly 60°. If even one angle is different, it's not equilateral.
• Using wrong area formula: The area is (√3/4)×side², not the standard (1/2)×base×height (though that formula also works if you calculate the height correctly).