How to Get Centroid Coordinates of a Right Triangle from Legs b and h

1.Legs are enough once you choose a placement

A right triangle is determined up to congruence by its two legs b and h (the sides that meet at the right angle). Coordinates need one more decision: where you put it. The homework convention is to sit the right angle at (0, 0), run b along the x-axis, and run h along the y-axis. After that choice, G has coordinates. Side lengths alone never give a unique (x, y) for G — you could translate or rotate the same triangle — so a problem that asks for coordinates is asking you to adopt this placement unless it already lists vertices. The hypotenuse c = √(b² + h²) is useful for a perimeter or a Pythagorean check; it does not appear in G.

2.Convert the legs into three vertices

Write A = (0, 0), B = (b, 0), and C = (0, h). Those are the only three points you need. If a prompt gives “base 6 and height 4” for a right triangle, it is naming the same two legs: b = 6 along the base, h = 4 as the altitude to that base from the right-angle setup. Do not park the hypotenuse on the x-axis and then treat those two numbers as intercepts — that is a different sketch. Why this origin is the one the shortcut uses is in How to Measure the Centroid from the Right-Angle Vertex.

\[ A=(0,0),\; B=(b,0),\; C=(0,h) \]

3.Average, or jump to (b/3, h/3)

The centroid is the arithmetic mean of the vertices: G = ((0+b+0)/3, (0+0+h)/3) = (b/3, h/3). You can stop at the shortcut once the vertices are in this form. Same result as typing the three points into the Centroid of a Triangle Calculator. The derivation of that collapse, and when you should not skip the average, is How the Right-Triangle Centroid Relates to the Three-Vertex Formula. The pillar walkthrough of this placement is How to Find the Centroid of a Right Triangle at (b/3, h/3).

\[ G=\left(\tfrac{b}{3},\tfrac{h}{3}\right) \]

4.Worked example: b = 6, h = 4

Vertices: (0, 0), (6, 0), (0, 4). Then Gₓ = (0+6+0)/3 = 2 and Gᵧ = (0+0+4)/3 = 4/3, so G = (2, 4/3) ≈ (2.00, 1.33). If the legs are 6 m and 4 m, G is at (2 m, 4/3 m) from the right angle; if they are 6 in and 4 in, the coordinates are in inches. Mixing units (one leg in cm, one in inches) makes G meaningless until you convert. Scaling both legs by k scales G by k: legs 12 and 8 send G to (4, 8/3). A 3-4-5 triangle with legs 3 and 4 is the same recipe at a smaller size: G = (1, 4/3). None of these needs the hypotenuse 5, 10, or √52.

\[ G=\left(\frac{0+6+0}{3},\frac{0+0+4}{3}\right)=\left(2,\tfrac{4}{3}\right) \]

5.Type the three vertices on the calculator

Open the Centroid of a Triangle Calculator and enter x₁ = 0, y₁ = 0, x₂ = 6, y₂ = 0, x₃ = 0, y₃ = 4. The outputs should match G x = 2.00 and G y = 1.33 (4/3). The site defaults are A(0, 0), B(6, 0), C(2, 4) — that C is not (0, 4), so those defaults are not this right triangle. If you leave C at (2, 4) you will not get (2, 4/3). Use the calculator when a later problem hands you three messy vertices instead of two legs; use (b/3, h/3) when the prompt really did give only b and h and the right angle is yours to place at the origin. To mark G on axes after you have the pair, see How to Plot the Centroid of a Right Triangle on the Coordinate Plane.

6.More right-triangle centroid guides

Why those coordinates are measured from the right-angle vertex: How to Measure the Centroid from the Right-Angle Vertex. The placement-first derivation: How to Find the Centroid of a Right Triangle at (b/3, h/3). When the average and the shortcut are the same identity: How the Right-Triangle Centroid Relates to the Three-Vertex Formula. If the right angle is already off the origin: How to Find the Centroid When the Right Angle Is Not at the Origin. A 2:1 median check: How to Check a Right-Triangle Centroid with Medians (2:1 Ratio).