Practice Problems for Special Right Triangles: 45-45-90 & 30-60-90

Special right triangles are right triangles whose side lengths follow fixed, memorizable ratios. The two you’ll meet constantly are the 45-45-90 triangle, with sides in the ratio 1 : 1 : √2, and the 30-60-90 triangle, with sides in the ratio 1 : √3 : 2. Working through practice problems builds the pattern recognition that lets you find missing sides in seconds instead of setting up the Pythagorean theorem every time. The sets below move from basic side-finding to multi-step figures, with worked solutions so you can check each step.

Warm-Up: Finding Missing Sides

In a 45-45-90 triangle the two legs are equal and the hypotenuse is √2 times a leg. In a 30-60-90 triangle, the shortest side (opposite 30°) is half the hypotenuse, and the longer leg (opposite 60°) is √3 times the short leg.

Problem 1. A 45-45-90 triangle has legs of 7 cm. Find the hypotenuse.

Problem 2. A 45-45-90 triangle has a hypotenuse of 12 in. Find each leg.

Problem 3. A 30-60-90 triangle has a short leg of 5 m. Find the other two sides.

Problem 4. A 30-60-90 triangle has a hypotenuse of 18 ft. Find both legs.

Worked Example

Here is Problem 2 in full. The hypotenuse equals leg × √2, so leg = 12 / √2. Rationalizing gives leg = 12√2 / 2 = 6√2 ≈ 8.49 in. You can verify with the Pythagorean theorem: (6√2)² + (6√2)² = 72 + 72 = 144, which is 12². Wolfram MathWorld lists these side ratios as standard right-triangle identities.

Practice: Working Backward and Scaling

Not every problem hands you a clean number. Practice scaling ratios up and down, and watch for the trap of confusing which side is the short leg.

Problem 5. A 30-60-90 triangle has a long leg of 9√3. What is the short leg, and what is the hypotenuse?

Problem 6. A 45-45-90 triangle has a leg of 4√5. What is the hypotenuse?

Problem 7. A 30-60-90 triangle has a short leg of 2.5. Give both remaining sides in exact form and rounded to two decimals.

Problem 8. The diagonal of a square is 10√2. What is the side length?

Solutions.

• 5. Long leg = short × √3, so short = 9√3 / √3 = 9, and the hypotenuse = 2 × 9 = 18.
• 6. Hypotenuse = 4√5 × √2 = 4√10.
• 7. Long leg = 2.5√3 ≈ 4.33; hypotenuse = 5.
• 8. A square’s diagonal splits it into two 45-45-90 triangles, so side = 10√2 / √2 = 10.

Problem 8 is really a 45-45-90 problem in disguise. Recognizing hidden special triangles is one of the biggest time-savers on tests. Khan Academy has a full unit on this skill if you want extra reps.

Mixed Figures and Multi-Step Problems

These combine the ratios with area, perimeter, or nested shapes. Draw the figure before you calculate anything.

Problem 9. An equilateral triangle has sides of 12. Find its height.

Problem 10. In a 30-60-90 triangle, the hypotenuse is 20. Find the area.

Problem 11. A rectangle is 6 by 6√3. What is the measure of the angle between its diagonal and its longer side?

Problem 12. A ladder leans against a wall at a 60° angle with the ground, reaching 15 ft up the wall. How long is the ladder?

Solutions.

• 9. An altitude on an equilateral triangle creates two 30-60-90 triangles with hypotenuse 12 and short leg 6, so height = 6√3 ≈ 10.39.
• 10. Short leg = 10, long leg = 10√3, area = ½ × 10 × 10√3 = 50√3 ≈ 86.6.
• 11. The diagonal forms a right triangle with legs 6 and 6√3. The ratio long leg : short leg = √3 : 1 matches a 60° angle, so the angle with the longer side is 30°.
• 12. The 15 ft side is opposite 60°, so it is the long leg. Short leg = 15 / √3 = 5√3 ≈ 8.66, and the hypotenuse (ladder) = 2 × 5√3 = 10√3 ≈ 17.32 ft.

A Word-Problem Strategy

For any applied problem, follow three steps:

1. Identify the right angle and mark which of the two special triangles it matches.
2. Label the known side as short leg, long leg, or hypotenuse.
3. Write the ratio equation and solve before rounding.

The National Council of Teachers of Mathematics emphasizes reasoning about structure rather than memorizing isolated formulas — exactly what labeling the triangle trains you to do.

Common Mistakes to Avoid

Most errors in these problems come from a few habits, and each is easy to fix.

• Treating √3 as 3. The long leg is √3 times the short leg, not 3 times it. If your answer is three times the short leg, you’ve made the swap.
• Halving the wrong side in a 30-60-90. Only the hypotenuse is twice the short leg. The long leg is never simply double anything.
• Forgetting to rationalize. Answers like 12 / √2 are usually marked incomplete; write 6√2 instead.
• Mixing up the two triangles. The 45-45-90 has two equal legs and √2 in the hypotenuse; the 30-60-90 has all three sides different and √3 in the longer leg.

A quick sanity check works well: the hypotenuse must always be the longest side, and in a 30-60-90 it is always exactly twice the shortest. If those two facts hold, the rest of your work was probably clean.

Deriving the Ratios Yourself

You don’t have to take the ratios on faith, and deriving them once makes them far harder to forget. For the 45-45-90 triangle, start with a square of side s. Drawing one diagonal splits it into two congruent right triangles, each with legs s and s. The Pythagorean theorem gives the hypotenuse as √(s² + s²) = √(2s²) = s√2. Since every such triangle is similar to this one, the 1 : 1 : √2 ratio holds at every scale.

For the 30-60-90 triangle, begin with an equilateral triangle of side 2. Drop an altitude from one vertex to the opposite side. This altitude bisects both the side and the 60° angle at the top, producing a right triangle with a hypotenuse of 2, a short leg of 1, and an acute angle of 30°. The altitude is the long leg: √(2² − 1²) = √3. That single construction gives you all three sides — 1, √3, and 2 — plus both acute angles.

Using the Ratios to Find Angles

The ratios also run backward: given two sides, you can identify the triangle. In a 45-45-90 triangle the legs are equal, so equal legs plus a right angle is a dead giveaway. In a 30-60-90 triangle the giveaway is the √3 relationship. If a right triangle has legs in the ratio 1 : √3, the angles opposite them are 30° and 60°. If a right triangle has legs in the ratio 1 : 1, both acute angles are 45°.

Problem 13. A right triangle has legs of 8 and 8√3. What are its acute angles?

Problem 14. A right triangle has a hypotenuse of 14 and one leg of 7. What are its acute angles?

Solutions. For 13, the ratio 8 : 8√3 simplifies to 1 : √3, so the angles are 30° and 60°. For 14, the leg is half the hypotenuse, which is the 30° signature — the opposite angle is 30°, and the remaining acute angle is 60°. Both checks rely only on the shape, not the specific size.

Estimating Without a Calculator

On tests without calculators, exact form is expected, but estimating keeps your answers honest. √2 ≈ 1.414 and √3 ≈ 1.732 are worth memorizing. So a 45-45-90 triangle with legs of 10 has a hypotenuse of about 14.14, and a 30-60-90 triangle with a short leg of 10 has a long leg of about 17.32 and a hypotenuse of 20.

Problem 15. A 30-60-90 triangle has a hypotenuse of 24. Estimate the perimeter.

Problem 16. A 45-45-90 triangle has a leg of 9. Estimate the perimeter.

Solutions. For 15, the short leg is 12 and the long leg is 12√3 ≈ 20.78, so the perimeter is about 12 + 20.78 + 24 ≈ 56.78. For 16, the hypotenuse is 9√2 ≈ 12.73, giving a perimeter of about 9 + 9 + 12.73 ≈ 30.73. Estimating first means a final exact answer that lands far from your estimate is a signal to recheck.

Where These Problems Show Up

The 45-45-90 and 30-60-90 ratios are not isolated exercises. They underlie the unit circle values you’ll meet in trigonometry, the diagonals of squares and cubes, and the heights of equilateral figures. Practicing them until the ratios are automatic gives you a shortcut that pays off across algebra, geometry, and precalculus.

If you want a structured sequence to keep drilling, the NCTM offers resources on geometric reasoning, and Khan Academy’s geometry course pairs each concept with graded exercises. The goal is not speed for its own sake — it is recognizing the ratio instantly so your attention can go to the harder part of a problem.

Key Takeaways

• A 45-45-90 triangle has sides in the ratio 1 : 1 : √2; a 30-60-90 has sides in the ratio 1 : √3 : 2.
• In a 30-60-90 triangle, the hypotenuse is twice the short leg, and the long leg is √3 times the short leg.
• Work backward by dividing: hypotenuse ÷ 2 gives the short leg, and long leg ÷ √3 gives the short leg.
• Many figures — squares, equilateral triangles, rectangles with √3 sides — hide a special right triangle inside them.
• Rationalize all denominators and keep answers in exact form unless the problem asks for a decimal.
• Always confirm the hypotenuse is the longest side; it catches most errors instantly.