Special Right Triangles Calculator
Special right triangles — the 45-45-90 triangle and the 30-60-90 triangle — are the two right triangles whose angles are fixed, which locks their side lengths into constant, predictable ratios. Because of that, you never need trigonometry to solve one: if you know just a single side, every other side follows automatically. This calculator solves both triangle types from any known side and returns exact radical answers alongside decimal approximations.
30-60-90 triangle: sides are in the ratio 1 : √3 : 2 → short leg 4 gives long leg 4√3 ≈ 6.93 and hypotenuse 8
Pick your triangle type and known side in the tool below — the formulas for each case are explained further down the page.
What Are Special Right Triangles
A right triangle has one 90-degree angle, and its two remaining angles always add up to 90 degrees. Most right triangles need trigonometry (sine, cosine, tangent) or the Pythagorean theorem to solve fully. But two specific right triangles — the 45-45-90 and the 30-60-90 — have fixed angle measures, which forces their side lengths into permanent, unchanging ratios. That's what makes them "special": once you know one side, the other two are immediate, no trig required.
- A 45-45-90 triangle is also called an isosceles right triangle, since its two legs are equal
- A 30-60-90 triangle is exactly half of an equilateral triangle, split down the middle
- Both triangles' side ratios come directly from the Pythagorean theorem, a² + b² = c²
- Knowing just one side length is enough to solve the entire triangle
The 45-45-90 Triangle
From a leg: hypotenuse = leg × √2
Example: leg = 5 → hypotenuse = 5√2 ≈ 7.07
From the hypotenuse: leg = hypotenuse ÷ √2
Example: hypotenuse = 10 → leg = 10 ÷ √2 = 5√2 ≈ 7.07
Both legs of a 45-45-90 triangle are always equal, since both non-right angles are equal (45° each). The hypotenuse is always the longer of the two side lengths — it's the leg length scaled up by √2 (approximately 1.4142).
The 30-60-90 Triangle
From the short leg (opposite 30°): long leg = short leg × √3, hypotenuse = short leg × 2
Example: short leg = 4 → long leg = 4√3 ≈ 6.93, hypotenuse = 8
From the hypotenuse: short leg = hypotenuse ÷ 2, long leg = short leg × √3
Example: hypotenuse = 8 → short leg = 4, long leg = 4√3 ≈ 6.93
In a 30-60-90 triangle, the shortest side is always opposite the smallest angle (30°), the longest leg is opposite the middle angle (60°), and the hypotenuse — always the longest side overall — is opposite the right angle (90°). The hypotenuse is exactly double the short leg, which is the fastest way to check your work.
Where the Ratios Come From
For the 45-45-90 triangle, set both legs to 1 and apply the Pythagorean theorem: 1² + 1² = c², so c² = 2, and c = √2 — that's the whole derivation. For the 30-60-90 triangle, start with an equilateral triangle with all sides equal to 2, then draw a line straight down from the top vertex to the midpoint of the base. That line splits the triangle into two 30-60-90 triangles, cuts the base into two segments of length 1 (the short leg), and the height of that line, by the Pythagorean theorem, works out to 1² + h² = 2², so h = √3 (the long leg) — with the original side of 2 remaining as the hypotenuse.
The Trap: Mixing Up the Short and Long Leg
Given a 30-60-90 triangle with long leg = 9, some solvers wrongly compute short leg = 9 ÷ 2 = 4.5
Correct: short leg = 9 ÷ √3 = 3√3 ≈ 5.196 — the ÷2 shortcut only works from the hypotenuse, never from the long leg
The most common error is applying the "divide by 2" shortcut to the wrong side. That shortcut only converts a hypotenuse into the short leg. If you're starting from the long leg instead, you must divide by √3, not by 2. Always double-check which side you were actually given before choosing a formula.
Real-Life Uses
- Construction and carpentry — roof pitches, staircases, and ramps frequently use 30-60-90 or 45-45-90 angles because their proportions are easy to mark and cut accurately
- Trigonometry and precalculus — these triangles are the source of the exact sine, cosine, and tangent values for 30°, 45°, and 60°, memorized and used throughout unit-circle work
- Navigation and surveying — quick distance and elevation estimates often rely on these fixed-angle triangles instead of a full trigonometric calculation
- Standardized tests — the SAT, ACT, and geometry exams test these ratios directly, since they can be solved without a calculator
- Design and architecture — diagonal bracing, gable ends, and tiling layouts commonly use one of these two angle sets
Glossary
- Hypotenuse
- The longest side of a right triangle, always located opposite the 90-degree angle.
- Leg
- Either of the two shorter sides of a right triangle that form the right angle.
- Isosceles Right Triangle
- Another name for the 45-45-90 triangle, since its two legs are equal in length.
- Short Leg / Long Leg (30-60-90)
- In a 30-60-90 triangle, the short leg is opposite the 30° angle and the long leg is opposite the 60° angle.
- Pythagorean Theorem
- The rule a² + b² = c², relating the two legs (a, b) and hypotenuse (c) of any right triangle.
Frequently Asked Questions
Q: What are the two types of special right triangles?
A: The 45-45-90 triangle (two 45° angles, sides in ratio 1:1:√2) and the 30-60-90 triangle (sides in ratio 1:√3:2).
Q: What is the 45-45-90 triangle rule?
A: The two legs are equal, and the hypotenuse equals a leg times √2. A leg of 5 gives a hypotenuse of 5√2 ≈ 7.07.
Q: What is the 30-60-90 triangle rule?
A: Sides follow the ratio 1 : √3 : 2. A short leg of 4 gives a long leg of 4√3 ≈ 6.93 and a hypotenuse of 8.
Q: How do you find the hypotenuse of a 45-45-90 triangle?
A: Multiply either leg by √2. A leg of 6 gives a hypotenuse of 6√2 ≈ 8.49.
Q: How do you find the sides of a 30-60-90 triangle from the hypotenuse?
A: Divide the hypotenuse by 2 to get the short leg, then multiply that by √3 to get the long leg. A hypotenuse of 10 gives a short leg of 5 and a long leg of 5√3 ≈ 8.66.
Q: Why are these triangles called "special"?
A: Their fixed angles lock in constant side ratios, so any side can be solved instantly from just one known side, without trigonometric tables.