Triangle Median Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The length of a triangle's median, connecting a vertex to the midpoint of the opposite side, is calculated with m = ½√(2b² + 2c² − a²), where a is the side opposite the starting vertex. For a right triangle with sides 3, 4, and 5, the median connecting the right-angle vertex to the midpoint of the hypotenuse is exactly 2.5 — half the hypotenuse.
Explanation
A median of a triangle connects one of its vertices to the midpoint of the opposite side; each triangle has exactly three medians (one per vertex), which all cross at a single point called the centroid (or center of gravity). This particular point divides each median in a constant 2:1 ratio, the longer portion always being on the vertex side. This calculator re-proves, through a notable special case, a property already met in our triangle circumradius calculator: for a right triangle, the median connecting the right-angle vertex to the midpoint of the hypotenuse is always exactly half the hypotenuse — a fact that is no coincidence, since the midpoint of the hypotenuse is precisely the center of the circle circumscribed about that right triangle, and this median is therefore nothing but a radius of that same circle. Another notable special case concerns the equilateral triangle, where each median is also an altitude and an angle bisector at once (a consequence of its perfect symmetry), of length (√3⁄2) × side — exactly what this calculator's first test case confirms, where an equilateral triangle with side 6 gives a median of about 5.196, identical to its altitude.
Example: right triangle with sides 3, 4, and 5, median to the hypotenuse
Inputs
Side a (hypotenuse, opposite side): 5 m. Side b: 4 m. Side c: 3 m.
Calculation
m = ½√(2×4² + 2×3² − 5²) = ½√(32 + 18 − 25) = ½√25 = ½ × 5 = 2.5 m.
Result
The median connecting the right-angle vertex to the midpoint of the hypotenuse is exactly 2.5 m.
Frequently asked questions
What is the difference between a median, an altitude, and an angle bisector?
The median connects a vertex to the geometric midpoint of the opposite side, the altitude connects a vertex to the opposite side forming a right angle with it, and the angle bisector connects a vertex to the opposite side dividing the vertex angle into two equal parts. These three segments coincide exactly only in an equilateral triangle (because of its perfect symmetry); in any other triangle, they are generally three distinct segments from the same vertex.
Where do the three medians of a triangle cross?
The three medians of a triangle always cross at a single point, called the centroid or center of gravity, which divides each median in a constant 2:1 ratio (two thirds of the median on the vertex side, one third on the opposite-side midpoint side). It is also the physical balance point of a triangular plate of uniform density.
What happens if the three entered lengths do not form a real triangle?
If the three lengths do not satisfy the triangle inequality (each side must be shorter than the sum of the other two), the term under the square root in the formula becomes negative, and this calculator shows an empty result rather than a meaningless number — exactly the same behavior as for a geometrically impossible triangle in our triangle area calculator (Heron's formula).