How it works
A 45-45-90 triangle is half of a square. Its legs are equal, and the hypotenuse is a leg times √2:
x : x : x√2
A 30-60-90 triangle is half of an equilateral triangle. The short leg (across from 30°) is half the hypotenuse, and the long leg (across from 60°) is the short leg times √3:
x : x√3 : 2x
Always start from the side you know and convert to the short leg first (for 30-60-90) or the leg (for 45-45-90); every other side follows. Two useful consequences: a square with side s has diagonal s√2, and an equilateral triangle with side s has height s√3 ÷ 2 and area s²√3 ÷ 4.
Special Right Triangles practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
A 45-45-90 triangle has legs of length 5. Give the hypotenuse in terms of √2 — enter the number before √2.
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Answer: 5√2
In a 45-45-90 triangle the hypotenuse is leg × √2, so it is 5√2.
A 30-60-90 triangle has long leg 9√3. Find the short leg.
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Answer: 9
The long leg is the short leg times √3, so the short leg is 9√3 ÷ √3 = 9.
A square has diagonal 16. Find its area.
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Answer: 128
The diagonal is side × √2, so the side is 16 ÷ √2 = 8√2. The area is (8√2)² = 64 × 2 = 128. Shortcut: a square's area is diagonal² ÷ 2 = 256 ÷ 2.
A 30-60-90 triangle has hypotenuse 20. Give the area in terms of √3 — enter the number before √3.
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Answer: 50√3
The short leg is half the hypotenuse, 10, and the long leg is 10√3. The legs are the base and height, so the area is 10 × 10√3 ÷ 2 = 50√3.
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Related topics
All Geometry topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice