How this instrument works
Given a triangle's three corners as plotted coordinates, each of its three edges has its own slope, found the ordinary way: rise over run between the two endpoints of that edge. This page computes all three at once — slope₁₂ between vertices 1 and 2, slope₂₃ between vertices 2 and 3, and slope₃₁ closing back from vertex 3 to vertex 1 — a genuinely different question from this site's Triangle Vertices page, which instead finds the enclosed area and perimeter from the identical kind of coordinate input.
A triangle's edge slopes reveal its shape in a way raw side lengths alone don't: two edges sharing the SAME slope would mean those two sides are parallel, which is geometrically impossible for a genuine triangle (any two of its three sides must eventually meet), so three plotted points with two matching edge slopes actually describe a degenerate, collapsed 'triangle' rather than a real one.
An isosceles triangle plotted with its base sitting horizontal shows a distinctive pattern in its edge slopes: the two slanted, equal-length sides carry slopes that are exact opposites of each other (one positive, one negative, same magnitude) — a direct visual and numerical signature of the mirror symmetry an isosceles triangle has around its own vertical axis.
- Enter the first corner's coordinates into the Vertex 1: x and Vertex 1: y fields.
- Enter the second corner's coordinates into the Vertex 2: x and Vertex 2: y fields.
- Enter the third corner's coordinates into the Vertex 3: x and Vertex 3: y fields.
- Read Slope of edge 1→2, Slope of edge 2→3, and Slope of edge 3→1: the sheet computes all three simultaneously.
Worked example — corners (0,0), (4,2), (2,6)
A triangle has corners at (0,0), (4,2), and (2,6). Its three edge slopes are 0.5, −2, and 3 — three completely different steepness values, one for each side, computed independently from the same set of three coordinate pairs.
A triangle with corners (0,0), (2,0), and (1,3) — an isosceles triangle sitting on a horizontal base — has edge slopes 0, −3, and 3: the horizontal base carries a slope of exactly 0, and the two equal slanted sides carry exactly opposite slopes, −3 and 3, a direct signature of the triangle's own left-right symmetry.
Questions
How do you find the slope of a triangle's edge?
Apply the ordinary slope formula, rise over run, to each pair of adjacent corners: (y₂−y₁)⁄(x₂−x₁) for the edge from vertex 1 to vertex 2, and the same pattern for the other two edges.
How is this different from the Triangle Vertices page?
That page finds the triangle's enclosed area and perimeter from its three corners. This page instead finds each individual EDGE's own slope — a question about the triangle's orientation and shape, not its size.
Can two edges of a triangle share the same slope?
No, not for a genuine triangle — two sides with matching slopes would be parallel, and parallel lines never meet, which would prevent the third corner from ever closing the shape. Three points producing two identical edge slopes describe a degenerate, collapsed figure instead.
What does it mean when two edge slopes are exact opposites?
It's the signature of an isosceles triangle plotted with its base horizontal — the two equal-length slanted sides mirror each other around the triangle's own vertical axis of symmetry, producing slopes of the same magnitude but opposite sign.
What if two vertices share the same x-coordinate?
The edge between them is vertical, and its slope is undefined, since the slope formula would divide by zero. A genuine triangle can have at most one such vertical edge, since two vertical edges would be parallel and could never meet at a third corner.