What is a Triangle Median?
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, which divides each median in a 2:1 ratio. This calculator computes the length of all three medians directly from the three side lengths a, b, and c.
How to Use the Calculator
Enter the three side lengths of your triangle into the fields labelled Side a, Side b, and Side c. Use any consistent unit (cm, m, inches — the result comes out in the same unit). Click calculate and you'll get the median to each side. The median ma is the one drawn to side a, mb to side b, and mc to side c.
The Formula Explained
The length of the median to side a is given by Apollonius's theorem:
$$m_a = \frac{1}{2}\sqrt{2b^{2} + 2c^{2} - a^{2}}$$
By symmetry, the other two medians swap the roles of the sides:
$$m_b = \frac{1}{2}\sqrt{2a^{2} + 2c^{2} - b^{2}} \quad \text{and} \quad m_c = \frac{1}{2}\sqrt{2a^{2} + 2b^{2} - c^{2}}$$
Notice that the median to side c uses the squares of sides a and b, NOT c — the side the median is drawn to is the one subtracted.
Worked Example
For a right triangle with sides a = 6, b = 8, c = 10:
$$m_c = \frac{1}{2}\sqrt{2\cdot6^{2} + 2\cdot8^{2} - 10^{2}} = \frac{1}{2}\sqrt{72 + 128 - 100} = \frac{1}{2}\sqrt{100} = 5.$$
$$m_a = \frac{1}{2}\sqrt{2\cdot64 + 2\cdot100 - 36} = \frac{1}{2}\sqrt{292} \approx 8.544.$$
$$m_b = \frac{1}{2}\sqrt{2\cdot36 + 2\cdot100 - 64} = \frac{1}{2}\sqrt{208} \approx 7.211.$$
FAQ
Do the three medians always meet at one point? Yes — they always intersect at the centroid, which is the triangle's center of mass.
What if my values don't form a valid triangle? The expression under the square root must be positive. For impossible side combinations the calculator returns 0.
Is the median the same as the altitude or angle bisector? No. A median goes to the midpoint of the opposite side, while an altitude is perpendicular to it and an angle bisector splits the angle. They coincide only in special cases like equilateral triangles.
