Equilateral Triangle Calculator

Calculate all properties of an equilateral triangle from the side length: area, perimeter, height, inradius, and circumradius.

Enter Side Length

Result

Area
43.30127
square units
Side Length (s) 10
Perimeter (P) 30
Height (h) 8.660254
Area (A) 43.30127
Inradius (r) 2.886751
Circumradius (R) 5.773503
Semi-perimeter (s) 15
Interior Angle 60 degrees

Step-by-Step Solution

A = (sqrt(3)/4) x s^2

What Is an Equilateral Triangle?

An equilateral triangle is a triangle in which all three sides are equal in length and all three interior angles are equal to 60 degrees. It is the most symmetric of all triangle types and is a special case of both isosceles and acute triangles. Due to its perfect symmetry, all properties of an equilateral triangle can be determined from just one measurement: the side length.

Equilateral Triangle Formulas

Area

The area enclosed by the equilateral triangle.

A = (sqrt(3)/4) * s^2

Perimeter

The total length of all three equal sides.

P = 3s

Height (Altitude)

The perpendicular distance from a vertex to the opposite side.

h = (sqrt(3)/2) * s

Inradius

The radius of the largest circle that fits inside the triangle.

r = s / (2*sqrt(3)) = s*sqrt(3)/6

Circumradius

The radius of the circle passing through all three vertices.

R = s / sqrt(3) = s*sqrt(3)/3

Relationship

The circumradius is always exactly twice the inradius.

R = 2r

Deriving the Area Formula

The area of an equilateral triangle can be derived using the general triangle area formula A = (1/2) x base x height. For an equilateral triangle with side s, the height h is found using the Pythagorean theorem on one of the two right triangles formed by the altitude:

  1. The altitude bisects the base, creating a right triangle with hypotenuse s and base s/2.
  2. By the Pythagorean theorem: h2 + (s/2)2 = s2
  3. h2 = s2 - s2/4 = 3s2/4
  4. h = s*sqrt(3)/2
  5. A = (1/2)(s)(s*sqrt(3)/2) = s2*sqrt(3)/4

Inradius and Circumradius Derivation

The inradius (apothem) r of an equilateral triangle can be found using the formula r = A/s_p, where A is the area and s_p is the semi-perimeter (3s/2). This gives r = (s2*sqrt(3)/4) / (3s/2) = s*sqrt(3)/6 = s / (2*sqrt(3)).

The circumradius R is found from the relationship between the circumradius, side length, and area of any triangle: R = (a*b*c) / (4A). For an equilateral triangle: R = s3 / (4 * s2*sqrt(3)/4) = s / sqrt(3) = s*sqrt(3)/3.

Practical Applications

  • Architecture: Equilateral triangles are used in truss designs, geodesic domes, and decorative patterns for their structural stability.
  • Engineering: Warning signs, yield signs, and many structural components use equilateral triangle geometry.
  • Nature: Crystal structures, molecular geometry, and snowflake patterns often exhibit equilateral triangle symmetry.
  • Tessellations: Equilateral triangles are one of only three regular polygons that can tile a plane perfectly.
  • Music: The triangle instrument and the triangular arrangement of notes in music theory.

Special Properties

  • All three angles are exactly 60 degrees.
  • The centroid, circumcenter, incenter, and orthocenter all coincide at the same point.
  • The altitude, median, angle bisector, and perpendicular bisector from any vertex are all the same line segment.
  • The circumradius is always exactly twice the inradius: R = 2r.
  • It has the maximum area among all triangles with a given perimeter.