Special Right Triangles Calculator
Enter any one side of a 45-45-90 or 30-60-90 triangle to find all other sides, area, and perimeter instantly.
📐 What are Special Right Triangles?
Special right triangles are right triangles whose angles produce side ratios that are exact, simple expressions. There are two classic special right triangles: the 45-45-90 triangle (the isosceles right triangle) and the 30-60-90 triangle. Because their angles are fixed, you can find all three sides from just one known measurement, using simple multiplication rather than the full Pythagorean theorem.
The 45-45-90 triangle has two equal 45-degree angles and a right angle. Its legs are equal in length, and the hypotenuse is the leg multiplied by the square root of 2 (approximately 1.41421). This triangle is everywhere in design and construction: it is the shape you get when you cut a square diagonally, it appears in 45-degree roof pitches, mitre cuts, and diagonal bracing in structural engineering. It is also the reference shape for the standard piano key layout.
The 30-60-90 triangle has angles of 30 degrees, 60 degrees, and 90 degrees. Its sides follow the ratio 1 : root3 : 2, meaning the short leg (opposite the 30-degree angle) is half the hypotenuse, and the long leg (opposite the 60-degree angle) equals the short leg multiplied by the square root of 3 (approximately 1.73205). This triangle is exactly half of an equilateral triangle, formed by drawing the altitude from one vertex to the midpoint of the opposite side. It appears in hexagonal geometry, equilateral grid layouts, and electrical engineering (three-phase AC systems use 120-degree spacing, which involves 30-60-90 geometry).
Both triangles are memorized by students for standardized tests because they allow fast mental calculations. The unit circle values for 30, 45, and 60 degrees all come directly from these two triangles. This calculator handles both triangle types in one place so you can switch between them without changing tools.
📐 Formulas
📖 How to Use This Calculator
Steps to Solve a Special Right Triangle
💡 Example Calculations
Example 1 - 45-45-90 Triangle from the Leg
A square room is 10 meters on each side. Find the diagonal.
Example 2 - 30-60-90 Triangle from the Hypotenuse
A rafter has a hypotenuse of 8 feet at a 30-degree pitch. Find the rise and run.
Example 3 - 45-45-90 Triangle from the Hypotenuse
A fence gate diagonal brace is 12 feet long. Find the frame width and height.
❓ Frequently Asked Questions
🔗 Related Calculators
What are the two special right triangles?
The two special right triangles are the 45-45-90 triangle (isosceles right triangle) and the 30-60-90 triangle. They are called special because their side lengths follow fixed ratios that allow you to find all sides from just one known measurement, without using the Pythagorean theorem directly.
What are the side ratios of a 45-45-90 triangle?
The sides of a 45-45-90 triangle are in the ratio 1 : 1 : root 2. Both legs are equal, and the hypotenuse is the leg multiplied by root 2 (approximately 1.41421). For example, if each leg is 5, the hypotenuse is 5 times root 2, which is approximately 7.071.
What are the side ratios of a 30-60-90 triangle?
The sides of a 30-60-90 triangle are in the ratio 1 : root3 : 2, corresponding to the short leg (opposite 30°) : long leg (opposite 60°) : hypotenuse (opposite 90°). If the short leg is a, the long leg is a times root3 and the hypotenuse is 2a.
How do you find the hypotenuse of a 45-45-90 triangle?
Multiply the leg by root 2 (approximately 1.41421). For example, if the leg is 8, the hypotenuse is 8 times root 2, which equals approximately 11.314. To find the leg from the hypotenuse, divide the hypotenuse by root 2.
How do you find the sides of a 30-60-90 triangle from the hypotenuse?
The short leg is always half the hypotenuse. The long leg is the short leg multiplied by root 3. For a hypotenuse of 10: short leg = 5, long leg = 5 times root 3 which equals approximately 8.660.
What is the area of a 45-45-90 triangle?
Area = half times leg squared = leg squared divided by 2. For example, if the leg is 6, the area is 6 squared divided by 2 = 36 divided by 2 = 18 square units. In terms of the hypotenuse c: Area = c squared divided by 4.
What is the area of a 30-60-90 triangle?
Area = half times short leg times long leg = half times a times a times root3 = a squared times root3 divided by 2. For a short leg of 4: area = 4 squared times root3 divided by 2 = 16 times 1.732 divided by 2 = approximately 13.856 square units.
Why are the 45-45-90 and 30-60-90 triangles called special?
They are called special because their angles are fixed and their side ratios are exact irrational numbers (involving root 2 or root 3). This makes them useful reference shapes in geometry, trigonometry, and engineering. Most other right triangles require the Pythagorean theorem and a calculator to solve; special right triangles can be solved with simple multiplication.
Where do special right triangles appear in real life?
Special right triangles appear in architecture (45-degree roof pitch, stair rise/run ratios), electrical engineering (AC voltage phasors), carpentry (45-degree mitre cuts), and construction. The 30-60-90 triangle appears in hexagonal layouts, equilateral grids, and structural trusses. Both appear extensively in trigonometry and unit circle calculations.
How is the 45-45-90 triangle related to a square?
A 45-45-90 triangle is exactly half of a square, formed by cutting a square diagonally. If the square has side length s, the resulting triangle has two legs of length s and a hypotenuse of s times root 2, which is the diagonal of the square.
How is the 30-60-90 triangle related to an equilateral triangle?
A 30-60-90 triangle is half of an equilateral triangle. When you draw the altitude of an equilateral triangle with side s, it divides the triangle into two 30-60-90 triangles, each with short leg s divided by 2, long leg s times root3 divided by 2 (the altitude), and hypotenuse s.
What are the trigonometric values for 30, 45, and 60 degrees?
From the 45-45-90 ratio: sin(45°) = cos(45°) = 1 divided by root2 = root2 divided by 2, and tan(45°) = 1. From the 30-60-90 ratio: sin(30°) = cos(60°) = 1 divided by 2, cos(30°) = sin(60°) = root3 divided by 2, tan(30°) = 1 divided by root3, tan(60°) = root3.