Centroid Triangle Calculator

Example data table

Formula used

How to use this calculator

1. Enter x and y coordinates for all three vertices.
2. Set your units label and preferred decimal rounding.
3. Enable steps if you want the workings displayed.
4. Click Submit to compute centroid, area, and medians.
5. Use the CSV or PDF buttons to export the report.

Centroid as a coordinate average

The centroid G is the balance point of a uniform triangular lamina. For vertices A(x₁,y₁), B(x₂,y₂), and C(x₃,y₃), the calculator computes x̄=(x₁+x₂+x₃)/3 and ȳ=(y₁+y₂+y₃)/3. This averaging property is numerically stable and makes G easy to verify by hand when inputs are simple.

Median intersection and the 2:1 rule

Each median connects a vertex to the midpoint of the opposite side. The three medians intersect at G, and the distance from a vertex to G equals two‑thirds of the full median length. The tool reports median lengths and vertex→centroid distances so you can validate geometric consistency across all three medians. In data entry, mismatched units typically break this 2:1 relationship immediately.

Area and collinearity diagnostics

To detect degenerate triangles, the calculator uses the shoelace determinant 2A = x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂). If |2A| is near zero, points are nearly collinear, area collapses, and centroid interpretation shifts from planar balance to a line‑segment average.

Perimeter and side length context

Side lengths a=|BC|, b=|AC|, and c=|AB| are computed with Euclidean distance √((Δx)²+(Δy)²). Perimeter helps compare triangles that share a centroid but differ in scale, and it supports quick reasonableness checks: moving one vertex far away should increase perimeter noticeably while pulling the centroid toward that vertex. If two sides are equal, the centroid still follows the same averaging rule.

Interpreting the plot for quality control

The interactive plot displays the triangle outline and centroid marker. Use it to spot swapped coordinates, sign errors, or unexpected orientation. A centroid outside the drawn triangle indicates an input or plotting issue, because for nondegenerate triangles G always lies strictly inside the region. Zooming in can reveal near‑collinear cases where the triangle appears almost flat.

Exportable results for reporting workflows

Engineering notes, classroom solutions, and QA logs often require reproducible outputs. The CSV export provides a tidy metric table for spreadsheets, while the PDF report captures vertices, centroid, area, and key distances. Together, they support traceable calculations across datasets, labs, and projects. Save consistent decimals to align with your organization’s rounding policy and audit expectations.

FAQs

1) Does the centroid depend on triangle orientation?

No. Rotating or reflecting the triangle changes vertex coordinates accordingly, but the centroid remains the average of the three vertices in that coordinate system.

2) What if my points are nearly collinear?

The centroid still computes, but the area becomes very small. Treat results as a near‑line case and consider improving coordinate precision or scaling.

3) Can I use negative coordinates?

Yes. The formulas work for any real coordinates, including negative x or y values, as long as all three vertices are provided.

4) Why show medians and vertex‑to‑centroid distances?

They provide cross‑checks. For each vertex, the distance to the centroid should be exactly two‑thirds of the corresponding median length.

5) Is the centroid always inside the triangle?

For any nondegenerate triangle, yes. If the plotted centroid appears outside, recheck entered coordinates or decimal rounding.

6) What do the CSV and PDF exports include?

They include vertices, centroid coordinates, area, perimeter, side lengths, and median metrics, making it easy to archive or share results.