A triangle is a polygon enclosed by three straight sides, and its angles can be acute, right, or obtuse. When acute and obtuse angles combine, the triangle is called oblique. By their sides, triangles are classified as scalene, isosceles, or equilateral. This guide explains the median of a triangle and the centroid where the medians meet.
- A median joins a triangle vertex to the midpoint of the opposite side, so every triangle has exactly three medians
- The three medians meet at the centroid, which divides each median in a 2:1 ratio from the vertex and acts as the triangle's centre of mass
- The median length is calculated from Apollonius's theorem.
What Is a Triangle and Its Area?
A triangle is a closed polygon bounded by three straight sides, with angles that may be acute, right, or obtuse. Its area is the flat space enclosed within those sides, measured in square units. For any triangle the area equals one-half the base multiplied by the perpendicular height, or altitude.
The area of a triangle is the interior space it occupies in two dimensions. It is always measured in square units, regardless of the triangle's shape. The formula for the area of a triangle is: Area = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex.
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The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, one from each vertex. In an equilateral triangle all three medians are equal in length, while in a scalene triangle no two medians are ever equal.
When a median is drawn from the vertex of an equal angle in an isosceles triangle, two of the medians share the same length. A median always lies inside the triangle. Each median splits the triangle into two smaller triangles of equal area, and together the three medians divide it into six smaller triangles of equal area.
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The centroid is the point where the three medians of a triangle intersect. It divides each median in a 2:1 ratio, lying two-thirds of the way from every vertex to the midpoint of the opposite side. The centroid is also the triangle's centre of mass, or balancing point.
Because the centroid divides each median in a 2:1 ratio measured from the vertex, it always sits inside the triangle. As the centre of mass, the centroid is the point at which a triangular sheet of uniform material would balance, which makes it important in engineering and physics.
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The median length is found using Apollonius's theorem, which states that the sum of the squares of two sides equals twice the square on the median to the third side plus twice the square on half that side. The median from vertex A equals one-half times the square root of 2b squared plus 2c squared minus a squared.
Written as a formula, the median to side a is: m = ½ × √(2b² + 2c² − a²). Here a, b and c are the lengths of the three sides, and m is the median drawn from the opposite vertex A to the midpoint of side a.
Example: A pizza is cut in the shape of a scalene triangle with corners labelled E, A and T, and you want to share it equally with a friend. Draw a median from any corner to the midpoint of the opposite side; this median splits the triangle into two parts of equal area, so the pizza is divided fairly.