Geometry · 6 min read
How to Get Centroid Coordinates of a Right Triangle from Legs b and h
Given only the two legs: build vertices (0, 0), (b, 0), (0, h), then G
When a problem gives only the two legs of a right triangle — not a sketch with plotted vertices — you still can report coordinates for G. Choose the usual placement (right angle at the origin, legs on the axes), write the three vertices from b and h, then average them (or use G = (b/3, h/3)). A worked case with b = 9 and h = 6 gives G = (3, 2). The hypotenuse is not an input.
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This placement assumes the right angle is at the origin and the legs lie along the axes. Side lengths alone do not pick a location in the plane until you choose that frame. The hypotenuse is not used.
1.Legs alone do not fix a point until you place the triangle
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.Turn the two 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 9 and height 6” for a right triangle, it is naming the same two legs: b = 9 along the base, h = 6 as the altitude in that 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 frame 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 the vertices, or use G = (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. 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 origin-formula walkthrough with a different worked size 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 from legs only: b = 9, h = 6
Vertices: (0, 0), (9, 0), (0, 6). Then Gₓ = (0+9+0)/3 = 3 and Gᵧ = (0+0+6)/3 = 2, so G = (3, 2). If the legs are 9 m and 6 m, G is at (3 m, 2 m) from the right angle; if they are 9 in and 6 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 18 and 12 send G to (6, 4). 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.
\[ G=\left(\frac{0+9+0}{3},\frac{0+0+6}{3}\right)=\left(3,2\right) \]
5.Check the three vertices on the calculator
Open the Centroid of a Triangle Calculator and enter x₁ = 0, y₁ = 0, x₂ = 9, y₂ = 0, x₃ = 0, y₃ = 6. The outputs should match G x = 3 and G y = 2. The site defaults are A(0, 0), B(6, 0), C(2, 4) — that C is not (0, 6), so those defaults are a general triangle, not this legs-only right triangle. Use the calculator when a later problem hands you three messy vertices; 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).
Try it yourself
Open the related calculator and put these formulas to work.