CMClearMathAcademy

Special Right Triangles

A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

Two right triangles appear so often that their side ratios are worth memorizing. A 45-45-90 (isosceles right) has both legs equal and a hypotenuse that is leg * sqrt(2). A 30-60-90 has short leg (opposite the 30 degree angle), long leg = short * sqrt(3), and hypotenuse = 2 * short. In Triangles, Similarity, and Trigonometry, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

What you'll learn

Why it matters: These ratios show up in carpentry (45 degree miter cuts), architecture (roof pitches), surveying, and the diagonal of a square (45-45-90).

Worked example

Problem. In a 30-60-90 triangle, the short leg is 5. Find the long leg and the hypotenuse.

  1. Long leg = short * sqrt(3) = 5 * sqrt(3).
  2. Hypotenuse = 2 * short = 10.
  3. Connect the result back to Special Right Triangles so the geometric relationship is explicit.
  4. Check the result against the original diagram, units, or proof statement.

Answer: Long leg = 5 * sqrt(3), hypotenuse = 10

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. In a 45-45-90 with both legs 6, find the hypotenuse coefficient in front of sqrt(2).

Show solution
  1. Hypotenuse = leg * sqrt(2) = 6 * sqrt(2).
  2. Coefficient is 6.

Answer: 6

2. In a 45-45-90, the hypotenuse is 10 * sqrt(2). Find one leg.

Show solution
  1. Hypotenuse = leg * sqrt(2), so leg = hypotenuse / sqrt(2) = 10 * sqrt(2) / sqrt(2) = 10.

Answer: 10

3. In a 30-60-90 with short leg 4, find the hypotenuse.

Show solution
  1. Hypotenuse = 2 * short leg = 2 * 4 = 8.

Answer: 8

4. In a 30-60-90 with short leg 4, find the long-leg coefficient in front of sqrt(3).

Show solution
  1. Long leg = short * sqrt(3) = 4 * sqrt(3).
  2. Coefficient is 4.

Answer: 4

5. In a 30-60-90 with hypotenuse 12, find the short leg.

Show solution
  1. Short leg = hypotenuse / 2 = 12 / 2 = 6.

Answer: 6

Practice this interactively with instant feedback and an AI tutor.

Practice Special Right Triangles Take the free placement check

More Geometry lessons