Calculate all three medians of a triangle from its side lengths. Find the centroid coordinates, area, median lengths, and reverse-solve sides from medians. Includes presets and reference table.
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 triangle's center of mass. The centroid divides each median in a 2 : 1 ratio from vertex to midpoint.
The median length formula (Apollonius's theorem) expresses each median in terms of the three sides: m_a = ½√(2b² + 2c² − a²). This elegant relationship means that knowing the three sides fully determines all three medians, and conversely, knowing all three medians lets you recover the side lengths.
Medians have numerous practical applications. In physics, the centroid is the balance point of a uniform triangular plate. In structural engineering, median lines help locate the center of gravity for triangular load-bearing elements. In computer graphics, the centroid is used for mesh simplification and barycentric coordinate systems.
This calculator works in both directions: enter three sides to get all medians, or enter three medians to recover the sides. It computes the centroid coordinates (when placed at a standard position), the area via Heron's formula, median-to-area relationships, and visual bars showing relative median lengths. Presets load classic triangles for quick exploration.
Triangle median problems often require several linked formulas at once: first validating the triangle, then applying Apollonius's theorem, then interpreting what the centroid and area results actually mean. This calculator keeps those relationships together so you can move from side data to medians, or from medians back to side lengths, without rebuilding the geometry each time.
It is especially useful when comparing special cases such as equilateral, isosceles, and right triangles, where the median lengths reveal structural symmetry. Teachers can use it to demonstrate centroid behavior, while students can verify reverse-solve exercises that are awkward to check by hand.
Median from vertex A: m_a = ½√(2b² + 2c² − a²) Median from vertex B: m_b = ½√(2a² + 2c² − b²) Median from vertex C: m_c = ½√(2a² + 2b² − c²) Centroid: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) Relation: m_a² + m_b² + m_c² = ¾(a² + b² + c²) Area from medians: A = (4/3)√[s_m(s_m−m_a)(s_m−m_b)(s_m−m_c)] where s_m = (m_a+m_b+m_c)/2
Result: m_a ≈ 4.27, m_b ≈ 3.61, m_c ≈ 2.50, Area = 6
For a 3-4-5 right triangle: m_a = ½√(2·16+2·25−9) = ½√73 ≈ 4.27, m_b = ½√(2·9+2·25−16) = ½√52 ≈ 3.61, m_c = ½√(2·9+2·16−25) = ½√25 = 2.5. The median to the hypotenuse is exactly half the hypotenuse.
A median is defined by position, not angle or perpendicularity. It always connects a vertex to the midpoint of the opposite side, which means every triangle has exactly three medians and they always intersect inside the figure. That makes medians more stable than altitudes in obtuse triangles and easier to use for centroid arguments.
The centroid is not just a geometric curiosity. It is the balancing point of a uniform triangular plate and appears whenever mass, load, or average position is involved. Because the centroid divides each median in a 2:1 ratio, median calculations help you locate it quickly in coordinate geometry, mechanics, and graphical modeling.
When solving by hand, first confirm the side lengths form a valid triangle. Then compute each median and compare their relative sizes to the opposite sides. In many common triangles, a symmetry check helps: equilateral triangles have equal medians, and right triangles have the special hypotenuse-median property. Those comparisons make it easier to spot arithmetic mistakes before moving on to centroid or area questions.
The centroid is the point where all three medians intersect. It is the center of mass (balance point) of a uniform triangular plate, located ⅔ of the way from each vertex along the median.
Use the median length formula: m_a = ½√(2b² + 2c² − a²), where a is the side opposite the vertex and b, c are the other two sides. Use this as a practical reminder before finalizing the result.
Yes. Using the inverse relationship: a = (2/3)√(2m_b² + 2m_c² − m_a²), and similarly for b and c.
The area can be found from medians using a Heron-like formula: A = (4/3)√[s_m(s_m−m_a)(s_m−m_b)(s_m−m_c)] where s_m is the semi-sum of medians. Keep this note short and outcome-focused for reuse.
Yes. Unlike altitudes (which can be outside for obtuse triangles), all three medians are always inside the triangle, connecting vertices to opposite midpoints.
In a right triangle, the median to the hypotenuse equals exactly half the hypotenuse length. This is because the midpoint of the hypotenuse is equidistant from all three vertices.