Median of Triangle Calculator
Enter the three side lengths of any triangle. The Median of Triangle Calculator instantly computes the length of all three medians using Apollonius’s theorem.
About the Median of Triangle Calculator
The Median of Triangle Calculator is a precise, scientifically validated tool that instantly computes the lengths of all three medians in any triangle using the proven Apollonius’s theorem (also known as the median length formula). This Median of Triangle Calculator uses the exact formula published in standard geometry textbooks (Coxeter & Greitzer 1967; Moise 1990) and is trusted by students, teachers, engineers, and mathematicians worldwide. Proudly supported by Agri Care Hub.
The Median Length Formula
m_b = ½ × √(2a² + 2c² − b²)
m_c = ½ × √(2a² + 2b² − c²)
Where a, b, c are the side lengths opposite vertices A, B, C respectively.
Why the Median of Triangle Calculator Is Important
A median is a line segment from a vertex to the midpoint of the opposite side. The three medians intersect at the centroid, which divides each median in a 2:1 ratio and is the triangle’s center of mass. Medians are fundamental in geometry, physics (moment of inertia), computer graphics (mesh processing), and engineering (structural analysis). Knowing median lengths is essential for solving advanced problems involving the centroid, area division, and triangle properties.
How to Use the Median of Triangle Calculator
- Enter the three side lengths of your triangle (any order is fine).
- Click “Calculate All Medians”.
- The tool instantly returns the length of each median with full mathematical steps.
When Should You Use This Calculator?
- High-school and college geometry courses
- Preparing for SAT, ACT, GCSE, IB, or university exams
- Teaching the centroid and median properties
- Engineering and physics problems involving mass distribution
- Computer graphics and 3D modeling (barycentric coordinates)
- Architectural design and structural analysis
Scientific Foundation
The median length formula was discovered by Apollonius of Perga (~200 BCE) and rigorously proven using vector geometry and the parallelogram law. It is equivalent to the distance from a vertex to the midpoint of the opposite side. The three medians always concur at the centroid, dividing each median in a 2:1 ratio (longer segment toward the vertex). This is one of the most elegant and useful theorems in plane geometry. Full treatment at Median of Triangle on Wikipedia.
Limitations
The calculator assumes a valid triangle (satisfies triangle inequality). Sides must be positive real numbers. Very large numbers may cause minor floating-point rounding.
Conclusion
The Median of Triangle Calculator brings one of the most beautiful theorems in geometry into an instant, user-friendly tool. Whether you’re a student mastering medians and the centroid, a teacher demonstrating proofs, or a professional working with structural balance, this calculator delivers mathematically perfect results every time. For more educational tools, visit Agri Care Hub.