Triangle Proportionality Theorem Calculator

Background.

Draw a line across a triangle parallel to one of its sides and it cuts the other two sides in the same proportion. If it meets one side 40% of the way down from the top corner, it meets the other side 40% of the way down as well, however different the two sides are in length. That is the triangle proportionality theorem, called the side-splitter theorem, the basic proportionality theorem or the intercept theorem depending on the textbook, and it appears as the second proposition of Book VI of Euclid's Elements.

The reason is similarity. The parallel line meets the two sides at the same angles the base does, so the small triangle it cuts off at the top has exactly the shape of the whole triangle, only smaller. Every length in the small triangle is the same fraction of the matching length in the large one, including the parallel segment itself, which is that fraction of the base.

Here you enter one full side, the part of it between the shared corner and the parallel line, and the full length of the other side. The page works out the scale ratio, part over whole, and applies it to the second side to give the matching part. The arithmetic is the same if both parts are measured from the base end instead, provided both are measured from the same end.

In practice that answers questions such as where a level brace meets the longer rafter of an uneven roof frame, or where a fence parallel to one boundary crosses the far edge of a triangular plot. The page takes the parallelism on trust: it cannot check that your line really is parallel, and the answer holds only if it is.

What is triangle proportionality theorem calculator?

The triangle proportionality theorem states that a line parallel to one side of a triangle, crossing the other two sides, divides those sides proportionally. In triangle ABC with D on AB, E on AC and DE parallel to BC, AD/AB = AE/AC, and equally AD/DB = AE/EC. The converse is also true: if two sides are divided in the same ratio, the line joining the division points is parallel to the third side. The shared ratio is the scale factor between the small triangle ADE and the whole triangle ABC.

How to use this calculator.

1. 01Identify the corner where the two cut sides meet, the one opposite the side your line runs parallel to. Enter the length of one cut side, from that corner to the base, as Full first triangle side.
2. 02Measure along that same side from the corner to where the parallel line crosses it, and enter it as Corresponding partial side. It must be shorter than the full side.
3. 03If you know the two pieces of the first side instead, add them for the full length; the piece touching the corner is the partial side.
4. 04Enter the Full second triangle side, from the same corner to the base, in the same unit as the first two lengths.
5. 05Read the corresponding partial side, measured from the same corner along the second side, and the scale ratio, which is also the length of the parallel segment as a fraction of the base.

The formula.

partialA / fullA = partialB / fullB, rearranged to partialB = (partialA ÷ fullA) × fullB. The first ratio is the scale factor k between the small triangle and the whole one. It lies between 0 and 1, which is why the partial side must be shorter than the full side. The theorem rests on similar triangles: because the cutting line is parallel to the base, it makes equal corresponding angles with both sides, so the small triangle has the same three angles as the large one and each of its sides is k times as long. The parallel segment is therefore k × the base, and the small triangle's area is k² × the area of the whole. The leftover pieces, fullA − partialA and fullB − partialB, are in the same ratio as well, which gives the theorem's other familiar form, AD/DB = AE/EC. All lengths must share one unit, and the result comes out in that unit.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.