Triangle Median Calculator
Calculate all three medians and find the centroid
Enter Triangle Sides
Median Formula
m_a = ½√(2b² + 2c² - a²)
Median from A to midpoint of side a
Triangle with Medians
Understanding Triangle Medians
A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has exactly three medians, and they all intersect at a single point called the centroid.
The Median Formula
For a triangle with sides a, b, and c, the median from vertex A to the midpoint of side a is:
The Centroid
The centroid (G) is the intersection point of all three medians. It has special properties:
- It divides each median in a 2:1 ratio from the vertex
- It's the center of mass (balance point) of the triangle
- Coordinates: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
2:1 Division Property
The centroid divides each median into two segments. The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.
Sum of Squared Medians
There's a relationship: m_a² + m_b² + m_c² = ¾(a² + b² + c²)
Frequently Asked Questions
What is a median of a triangle?
A median connects a vertex to the midpoint of the opposite side. Every triangle has three medians.
Where do the medians intersect?
All three medians intersect at a single point called the centroid (G). This is true for every triangle.
What does the 2:1 ratio mean?
The centroid divides each median so that the vertex-to-centroid segment is twice as long as the centroid-to-midpoint segment.
Why is the centroid called the center of mass?
If you cut a triangle from uniform material, it will balance perfectly on a point placed at the centroid.
Is the centroid always inside the triangle?
Yes! Unlike the circumcenter or orthocenter, the centroid is always located inside the triangle, regardless of the triangle type.
How do I find the centroid from coordinates?
If vertices are (x₁,y₁), (x₂,y₂), (x₃,y₃), the centroid is at ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). Simply average the coordinates.
How does a median differ from an altitude?
A median goes to the midpoint of a side; an altitude is perpendicular to a side. They're the same only for isosceles triangles (to the base).
What is the medial triangle?
The medial triangle is formed by connecting the three midpoints. It's similar to the original with a scale factor of 1/2.