AI Tutor
What if you could solve an entire right triangle in your head in two seconds—without ever squaring numbers or calculating square roots?
By slicing two of geometry's most basic shapes—a square and an equilateral triangle—in half, ancient Greek geometers uncovered two special triangles with fixed, predictable side ratios.
In any right triangle, the legs are the two sides that form the 90∘90^\circ right angle, and the hypotenuse is the longest side directly opposite the 90∘90^\circ angle.
The Pythagorean theorem proves that the sum of the squares of the legs equals the square of the hypotenuse: a2+b2=c2a^2 + b^2 = c^2
What happens when both legs of a right triangle are the exact same length? Let's slice a square in half to find out.

The 45°-45°-90° Triangle

When you draw a diagonal across a square of side length xx, you divide the square into two identical triangles with angle measures of 45∘45^\circ, 45∘45^\circ, and 90∘90^\circ.
Because the legs both equal xx, the Pythagorean theorem gives x2+x2=c2x^2 + x^2 = c^2, which simplifies to 2x2=c22x^2 = c^2, meaning the hypotenuse is always x2x\sqrt{2}.
This creates the permanent ratio x:x:x2x : x : x\sqrt{2} for every 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle.
If a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle has a leg of 6 cm6\text{ cm}, its other leg is 6 cm6\text{ cm} and its hypotenuse is instantly 62 cm6\sqrt{2}\text{ cm}.
Squares give us 45∘45^\circ-45∘45^\circ-90∘90^\circ triangles—but what pattern emerges if we slice an equilateral triangle straight down the middle?

The 30°-60°-90° Triangle