Special Right Triangles (30-60-90, 45-45-90)

Geometry · Axiom Academy

LESSON Special Right Triangles (30-60-90, 45-45-90) Master the 45-45-90 and 30-60-90 triangles — and find missing sides without the Pythagorean theorem every time. The 45-45-90 is an isosceles right triangle : its two legs are equal, and the two non-right angles are both 45°. Cut a square along its diagonal and you get exactly this triangle — the diagonal becomes the hypotenuse. Side ratio (leg : leg : hypotenuse) The hypotenuse is a leg times √2 A 45-45-90 triangle has legs of length 6. Find the hypotenuse. The 30-60-90 is half of an equilateral triangle . Drop an altitude from the top vertex of an equilateral triangle and it splits into two of these. The base gets halved — so the short leg (opposite 30°) is always half the hypotenuse, and the altitude becomes the long leg (opposite 60°). Length x — the smallest side, always half the hypotenuse. Length x√3 — the short leg scaled by √3. Length 2x — exactly twice the short leg. short : long : hyp = 1 : √3 : 2. A 30-60-90 triangle has a hypotenuse of 10. Find both legs. Short leg = 5, Long leg = 5√3 ≈ 8.66 Both ratios work for any triangle with these angles — scale the whole triangle and the ratio holds. Here are the two patterns side by side. 45-45-90: two equal legs, hypotenuse = leg × √2. 30-60-90: short leg opposite 30°, long leg opposite 60°. In 30-60-90, the hypotenuse is always twice the short leg. In 30-60-90, the long leg = short leg × √3. These ratios work for any size triangle with these angles.

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