Thales' Theorem Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
Thales' theorem gives AN = (AM × AC) ÷ AB and MN = (AM × BC) ÷ AB, when (MN) is parallel to (BC) in a triangle ABC. For AM=3, AB=5, AC=7, and BC=6, this gives AN=4.2 and MN=3.6.
Explanation
Thales' theorem, one of the pillars of geometry taught in secondary school, applies whenever a line (MN) is drawn parallel to a side (BC) of a triangle ABC, cutting the other two sides at M (on [AB]) and N (on [AC]). It then establishes a proportionality between three length ratios: AM/AB = AN/AC = MN/BC — in other words, the small triangle AMN is a scaled-down (or scaled-up) version of the large triangle ABC, with the same scale ratio on all three sides. This calculator uses this equality of ratios to derive the lengths AN and MN from the three other known measurements (AM, AB, AC, and BC), without needing to know the triangle's angles or draw the figure. This theorem has many practical applications beyond the classroom exercise: it can, for example, estimate the height of an object too tall to measure directly (a tree, a building) from the shadow it casts, compared to that of a reference object of known height — this is in fact the method by which Thales himself is said to have measured the height of the Great Pyramid of Giza in antiquity, according to the tradition reported by Greek historians.
Example: a triangle with AM=3, AB=5, AC=7, BC=6
Inputs
AM = 3. AB = 5. AC = 7. BC = 6.
Calculation
AN = (3 × 7) ÷ 5 = 21 ÷ 5 = 4.2. MN = (3 × 6) ÷ 5 = 18 ÷ 5 = 3.6.
Result
In this triangle, AN measures 4.2 and MN measures 3.6 — the small triangle AMN is a reduction of the large triangle ABC at a ratio of 3/5 (0.6) on each of its sides.
Frequently asked questions
Does Thales' theorem work if (MN) isn't parallel to (BC)?
No, the parallelism between (MN) and (BC) is the essential condition for this equality of ratios to hold — without it, there's no simple proportionality relationship between the lengths of the two triangles. It's actually the converse of Thales' theorem that, conversely, lets you prove two lines are parallel by checking that the relevant length ratios are indeed equal.
How would Thales have measured the height of the pyramid of Giza?
According to the tradition reported by ancient historians, Thales is said to have planted a vertical stick of known height next to the pyramid, and compared the length of its shadow to that of the pyramid at the same moment. Since the sun's rays are nearly parallel at this scale, the two shadows and the two heights form similar triangles: the ratio between the stick's height and its shadow is the same as the ratio between the pyramid's height and its shadow, allowing the latter to be deduced without having to climb it — a proportional reasoning also found in our rule of three calculator, for more general situations.
What happens if M coincides exactly with B?
In that case, illustrated by this calculator's third test case, the line (MN) coincides exactly with (BC), and N then necessarily coincides with C — the small triangle AMN becomes identical to the large triangle ABC, with a scale ratio of 1. This is an edge case that confirms the formula's consistency: AM=AB does indeed give AN=AC and MN=BC. This theorem is distinct from the Pythagorean theorem, which relates the side lengths of a single right triangle rather than two similar triangles.