Centroid of a Triangle: the vertex-average formula, the 2:1 median split, and where it sits among the other triangle centres

What the centroid of a triangle is

The centroid of a triangle is the single point where the triangle's three medians meet. A median is the straight line segment joining a vertex to the midpoint of the opposite side, and a triangle has exactly three of them — one per vertex. Those three segments are guaranteed to cross at one common point, and that point is the centroid. The centroid of a triangle calculator takes the three vertex coordinates and returns that point, plus the side midpoints, the lengths of the three medians, and the triangle area for cross-checking.

The same point shows up in physics as the centre of mass of a triangular plate of uniform density. If you cut a triangle out of cardboard and try to balance it on a pencil tip, the balance point is the centroid. That dual identity — geometric median intersection and physical centre of mass — is why the centroid is sometimes called the "centre of gravity" or "barycentre" of the triangle, and it is the most useful triangle centre to keep in your head if you are working in computer graphics, structural engineering, or any field that decomposes shapes into triangles.

The formula in one line

For a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the centroid is:

G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)

That is the entire computation. Take the arithmetic mean of the three x-coordinates to get Gx, take the arithmetic mean of the three y-coordinates to get Gy, and you have the centroid. No trigonometry, no square roots, no special cases for acute versus obtuse triangles. The same recipe extends to triangles in three-dimensional space by averaging the z-coordinates as well, and to higher-dimensional simplices by averaging all their vertex components in parallel. The centroid of a triangle calculator runs this directly in 2D and returns the point along with the intermediate midpoints so you can see the geometry behind the numbers.

Why averaging the vertices is the right answer

There are two ways to see why the centroid is the average of the vertices, and they are both worth carrying. The first is physical: model the triangle as three equal point masses at the vertices. The centre of mass of any collection of point masses is ΣmᵢVᵢ / Σmᵢ, and with three equal masses this collapses to (V₁ + V₂ + V₃) / 3 — the unweighted average. That argument tells you the average-of-vertices formula is the centre of mass for three equal masses; a separate argument (integrating over the triangle) shows that the centre of mass of a uniform-density triangular plate is the same point. Both roads lead to the same coordinates.

The second is geometric: prove the centroid is two-thirds of the way from each vertex to the midpoint of the opposite side, using vector arithmetic. Pick vertex V₁ and let M₁ be the midpoint of the side opposite V₁, so M₁ = (V₂ + V₃) / 2. Define G = (V₁ + V₂ + V₃) / 3 and look at G − V₁:

G − V₁ = (V₁ + V₂ + V₃)/3 − V₁ = (V₂ + V₃ − 2V₁) / 3 = (2/3) · ((V₂ + V₃)/2 − V₁) = (2/3) · (M₁ − V₁).

That last equality says G is exactly two-thirds of the way from V₁ to M₁, which means it lies on the median V₁M₁. The argument is symmetric in the three vertices, so the same point lies on the median from V₂ and the median from V₃ as well — proving in one line that the three medians are concurrent and that their common point is the average of the vertices.

Worked example: A = (0, 0), B = (6, 0), C = (3, 6)

These are the default values the centroid of a triangle calculator starts with — a clean isoceles triangle sitting in the first quadrant. Run the formula step by step:

1. Sum the x-coordinates. 0 + 6 + 3 = 9.
2. Sum the y-coordinates. 0 + 0 + 6 = 6.
3. Divide each sum by 3. Gx = 9 / 3 = 3, Gy = 6 / 3 = 2.
4. Read the centroid. G = (3, 2).
5. Sanity-check via the medians. The midpoint of BC, opposite A, is ((6 + 3)/2, (0 + 6)/2) = (4.5, 3). The median from A runs from (0, 0) to (4.5, 3); travel two-thirds of the way and you land at (3, 2), the centroid. The same two-thirds rule works from B and from C, and all three medians cross at (3, 2).
6. Cross-check with the area. The shoelace area of A, B, C is |0·(0 − 6) + 6·(6 − 0) + 3·(0 − 0)| / 2 = 36 / 2 = 18 square units. A non-zero area confirms the three points are not collinear, so the centroid is well-defined.

Type the same vertices into the calculator and the headline result is the centroid (3, 2); the breakdown also reports the three side midpoints — (4.5, 3) opposite A, (1.5, 3) opposite B, (3, 0) opposite C — and the three median lengths via the Euclidean distance from each vertex to its opposite midpoint.

The 2:1 median ratio and what it means in practice

The vector proof above shows that the centroid lies two-thirds of the way from each vertex to the opposite side's midpoint. In other words, on every median, the segment from the vertex to the centroid is exactly twice as long as the segment from the centroid to the midpoint. The 2:1 split is one of the most cited facts about triangles and shows up whenever a structure is balanced on a single triangular support.

Two consequences are worth flagging. The first is the median length formula: each median has length proportional to the full segment from vertex to opposite midpoint, and the Euclidean distance between those two points is what the distance calculator runs. The second is that the centroid divides the triangle into three smaller triangles (each with one vertex at the centroid and one full side of the original) of equal area. The 2:1 split on the medians forces each of those three regions to be exactly one third of the total triangle area, which is the cleanest way to see why the centroid is the balance point without invoking any physics.

The four classical triangle centres compared

The centroid is one of four classical "triangle centres". The other three answer different questions about the same triangle, and only in the equilateral case do all four coincide at one point.

Centroid

Average of the vertices. Centre of mass for a uniform-density triangle. Always inside the triangle. Found by intersecting the three medians. This is the one the centroid calculator computes.

Incentre

The point equidistant from the three sides — the centre of the inscribed circle (the largest circle that fits inside the triangle). Found by intersecting the three angle bisectors. Always inside the triangle. Coordinates are a weighted average of the vertices, where the weights are the lengths of the opposite sides.

Circumcentre

The point equidistant from the three vertices — the centre of the circumscribed circle (the unique circle that passes through all three vertices). Found by intersecting the three perpendicular bisectors of the sides. Sits inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle.

Orthocentre

The point where the three altitudes meet (an altitude is the perpendicular from a vertex to the opposite side, or to its extension). Sits inside an acute triangle, at the right-angle vertex of a right triangle, and outside an obtuse triangle.

For any triangle that is not equilateral, the centroid, circumcentre, and orthocentre are distinct but always collinear — they lie on a single straight line called the Euler line, with the centroid sitting one-third of the way from the circumcentre to the orthocentre. The incentre, in contrast, only joins the Euler line when the triangle is isoceles, so it is the odd one out. The general triangle calculator focuses on side and angle relationships rather than centres, but pairing it with the centroid tool is the cleanest way to get all the standard triangle facts in one session.

Common mistakes

• Confusing the centroid with the incentre or circumcentre. All three are "centres", but only the centroid is the unweighted average of the vertices. The incentre is the weighted average using opposite side lengths, and the circumcentre comes from the perpendicular bisectors. Plugging one formula in expecting another's answer is the most common beginner mistake — if your "centroid" sits outside the triangle, something has gone wrong, because the real centroid never does.
• Averaging midpoints instead of vertices. The centroid is the average of the three vertices, not the average of the three side midpoints. Averaging the midpoints happens to give the same answer (because each midpoint is itself the average of two vertices, and averaging averages folds back to the same point), but it is the slow road and tends to introduce arithmetic slips. Stick with the vertices.
• Forgetting to divide by 3. The sum (x₁ + x₂ + x₃) is three times the centroid's x-coordinate, not the x-coordinate itself. Forgetting the division by 3 is the algebraic equivalent of forgetting the ½ in the ½·base·height area formula — your answer is off by a factor of 3 and only the dimensional sanity check tends to catch it.
• Using collinear inputs. If your three points lie on a single straight line, there is no triangle and no centroid. The calculator checks the shoelace area before computing the centroid and flags the input as degenerate when the area is zero. If you see the "vertices are collinear" message, move one of the three points off the line to recover a proper triangle.
• Mixing coordinate systems. Vertices have to live in the same coordinate frame. If two are in metres and one is in feet, the centroid is mathematically well-defined but physically meaningless. Convert all three to the same units before typing them in.

Where the centroid actually gets used

The centroid is the workhorse triangle centre because the formula is cheap and the geometry is well-behaved (always inside, always the centre of mass, always the average of the vertices). In computer graphics, the centroid is the natural anchor for affine transformations of a triangle — rotate or scale around the centroid and the triangle stays balanced around its visual centre. In finite element analysis, the centroid is the integration point for first-order quadrature on triangular elements, which is why every textbook treatment of triangular meshes uses the same average-of-vertices formula. In structural engineering, the centroid of a cross-section determines the neutral axis for bending — beams pivot through the centroid, not through the geometric "middle" by eye. In GIS and surveying, the centroid of a triangulated patch is the simplest summary statistic for the patch's location, and it is what most software returns when you ask for "the centre" of an irregular polygon decomposed into triangles.

The centroid also shows up as a coordinate convenience: if you have three points and you want a single representative location, the centroid is the obvious choice because it respects every vertex equally. The distance calculator and the slope calculator handle the downstream questions — how far is each vertex from the centroid, and along what bearings — once you have the centroid coordinates in hand.

When the centroid is not what you want

Two situations push you toward a different triangle centre. If you need the point equidistant from the three sides — for instance, the centre of the largest inscribed circle or the point that minimises maximum distance to the sides — you want the incentre, which is a weighted average of the vertices using opposite side lengths, not the unweighted average. If you need the point equidistant from the three vertices — for instance, the centre of a circumscribing circle in a surveying or sensor-placement problem — you want the circumcentre, which comes from the perpendicular bisectors of the sides. The centroid is the right answer when you want the centre of mass or the balance point, and the wrong answer for these other two problems.

There is also a subtlety around weighted centres of mass. If your three vertices carry unequal masses, the centre of mass is the weighted average ΣmᵢVᵢ / Σmᵢ rather than the centroid (V₁ + V₂ + V₃) / 3. The centroid uses equal weights by definition; if your physical model has unequal masses at the vertices, plug them into the weighted formula yourself rather than expecting the centroid calculator to do it.

Related calculators and where to go next

The centroid of a triangle calculator is the direct tool for the average-of-vertices formula. For broader triangle work — side lengths, angles, classification — the general triangle calculator handles SSS, SAS, ASA, and AAS data in one interface. If you need the area of a triangle from two sides and the included angle, the area of an oblique triangle calculator runs ½·a·b·sin(C) and also returns the third side and the remaining angles. For classifying a triangle as acute, right, or obtuse from three sides alone, the acute triangle calculator does the sharpened Pythagorean test. And for coordinate-level operations downstream of the centroid, the distance calculator and the slope calculator handle the single-pair-of-points questions one step at a time.

Frequently asked questions

See the FAQ section below the calculator on the centroid of a triangle calculator page for direct answers to the questions that come up most often — what the centroid is, why the formula is the average of the vertices, why the 2:1 median ratio holds, how the centroid differs from the incentre and circumcentre, what to do with collinear inputs, and how the formula extends to three-dimensional triangles. The same FAQ items are marked up with FAQPage schema so search engines can surface them directly.