A ternary diagram displays three variables inside an equilateral triangle, where every point represents a mixture that adds up to 100%. Each corner stands for 100% of one component, and the opposite side represents 0% of that same component. Once you understand that single rule, the rest follows logically.
The Basic Geometry
The triangle has three corners (called apices), and each one is labeled with a component. In a soil texture triangle, for example, the three corners are sand, silt, and clay. In a rock classification chart, they might be quartz, alkali feldspar, and plagioclase. Regardless of the subject, the structure works the same way.
A point sitting right at a corner means the sample is 100% that single component. A point along one side of the triangle means the sample contains only two of the three components, with none of the third. Any point inside the triangle contains all three.
The key constraint: the three percentages always sum to 100%. If a sample is 40% A and 35% B, it must be 25% C. There is no wiggle room. This is what makes the triangle work as a two-dimensional plot for three variables.
How the Grid Lines Work
Most ternary diagrams include a grid of evenly spaced lines, usually at 10% or 20% intervals. These grid lines run parallel to each side of the triangle. That parallel relationship is the single most important thing to grasp, because it tells you which lines belong to which component.
Here is the rule: the set of grid lines for a given component runs parallel to the side of the triangle that is opposite that component’s corner. So if component A sits at the top apex, its grid lines run parallel to the bottom side. The line closest to the bottom side is 0% A, and the line closest to the top corner is 100% A. Each component’s percentage increases as you move toward its corner and decreases as you move away.
A common layout places one component at the top and the other two at the bottom left and bottom right. In this arrangement, the percentage scales typically rotate around the triangle. The right side shows percentages for the top component, the bottom shows percentages for the lower-right component, and the left side shows percentages for the lower-left component. The direction of increase (clockwise or counterclockwise) depends on how the diagram’s creator chose to label it, so always check the tick marks before reading values.
Reading a Point Step by Step
Say you have a point plotted inside the triangle and you want to find its three percentages. Pick one component to start with. From the point, draw (or imagine) a line that runs parallel to the side opposite that component’s corner. Where this line hits the labeled scale on one of the other two sides, read off the percentage. Repeat for a second component using the same method, drawing a line parallel to the side opposite its corner. You can find the third component by subtracting the first two from 100%, or you can verify by drawing all three lines.
For example, imagine a triangle with component A at the top, B at the lower left, and C at the lower right. To find the percentage of A, draw a horizontal line (parallel to the base) through your point. If it crosses the right-side scale at 30, the sample is 30% A. To find the percentage of B, draw a line through the point parallel to the side connecting A and C. If it crosses the left-side scale at 50, the sample is 50% B. Component C is then 100 minus 30 minus 50, which equals 20%.
If the diagram has a reference grid already printed on it, you can skip the line-drawing and simply follow the nearest grid lines to read values directly. Just make sure you are following the correct set of parallel lines for each component.
A Common Mistake to Avoid
The most frequent error is following the wrong set of grid lines. Because three families of parallel lines overlap inside the triangle, it is easy to accidentally trace a line that belongs to the wrong component. Before reading any values, identify which direction each component’s lines run. Confirm by checking that the lines for component A are parallel to the side opposite A’s corner. If your three readings do not add up to 100%, you have likely mixed up the grid lines.
Another pitfall shows up when the diagram’s sides are not labeled with numbers. In that case, count the tick marks along each edge. Most diagrams use 10% increments, giving you 10 intervals per side. If the tick marks are present but the numbers are missing, you can reconstruct the scale by connecting tick marks across the triangle to build your own grid.
Normalizing Your Data Before Plotting
Ternary diagrams require that the three plotted values add up to exactly 100%. If your raw data includes other components not shown on the diagram, you need to normalize first. This means recalculating each of the three components as a percentage of their combined total, ignoring everything else.
Suppose a rock sample contains 25% quartz, 35% alkali feldspar, 15% plagioclase, and 25% other minerals. The three components shown on the diagram total 75%. To normalize: quartz becomes 25/75 = 33.3%, alkali feldspar becomes 35/75 = 46.7%, and plagioclase becomes 15/75 = 20.0%. These three now sum to 100% and can be plotted.
The Soil Texture Triangle
One of the most widely used ternary diagrams is the USDA soil texture triangle. It plots sand (particles 2.0 to 0.05 mm), silt (0.05 to 0.002 mm), and clay (smaller than 0.002 mm). The triangle is divided into named regions like “sandy loam,” “silty clay,” and “loam,” each defined by specific boundary percentages.
To use it, locate the percentage of clay on the clay axis and follow the line parallel to the base of the triangle. Then locate the percentage of sand on the sand axis and follow that line parallel to the appropriate side. Where the two lines cross falls inside one of the named texture classes. You can double-check by tracing the silt percentage the same way. For instance, a soil with 20% clay, 40% silt, and 40% sand lands in the “loam” region.
Rock Classification Diagrams
Geologists use the QAPF diagram to classify igneous rocks. The upper triangle plots quartz (Q), alkali feldspar (A), and plagioclase (P). The horizontal lines represent the percentage of quartz relative to the total of all three minerals, while the inclined and vertical lines represent the ratio of plagioclase to the combined alkali feldspar plus plagioclase. The intersecting boundaries define rock names like granite, granodiorite, and tonalite.
The reading method is the same parallel-line technique, but the labeled boundaries do the interpretation for you. Once you plot your normalized mineral percentages, the region your point falls in tells you the rock’s official classification.
Phase Diagrams and Tie Lines
In chemistry and materials science, ternary diagrams sometimes include curved boundaries and straight lines called tie lines. A curved boundary (often called a binodal curve) separates a single-phase region from a two-phase region. Inside the two-phase region, any composition will split into two distinct phases.
Tie lines connect the compositions of those two coexisting phases. If your overall mixture plots at a point along a tie line, the two endpoints of that line tell you the composition of each phase. The closer your point is to one endpoint, the more of that phase is present in the mixture. This relationship is quantified by the lever rule: the fraction of each phase is proportional to the distance from your point to the opposite endpoint, divided by the total length of the tie line.
Tie lines within the same two-phase region never cross each other. Their slope can change gradually across the region, which reflects how the partitioning of components between the two phases shifts as the overall composition changes.
Plotting a Point From Scratch
If you need to place a point on a blank or partially labeled ternary diagram, start by identifying which corner belongs to which component and confirming the direction of increasing percentage. Then take your first component’s value, find that percentage on its scale, and draw a faint line through the triangle parallel to the side opposite its corner. Do the same for a second component. The intersection of those two lines is your point. The third component’s value is automatically satisfied because the geometry enforces the 100% constraint.
For accuracy, use a straightedge and align it carefully with the tick marks. Even a small angular error can shift your point noticeably, especially in diagrams with tightly spaced classification boundaries like the soil texture triangle.