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.

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

45-45-90: a = b | c = a × √2

a, b = the two equal legs (both opposite 45°)

c = hypotenuse (opposite 90°)

Ratio: 1 : 1 : √2 = 1 : 1 : 1.41421...

Area = a² ÷ 2

Perimeter = 2a + a√2 = a(2 + √2)

Example: leg = 7 → hypotenuse = 7 × √2 = 9.899; area = 49 ÷ 2 = 24.5

30-60-90: b = a√3 | c = 2a

a = short leg (opposite 30°) = hypotenuse ÷ 2

b = long leg (opposite 60°) = a × √3

c = hypotenuse (opposite 90°) = 2a

Ratio: 1 : √3 : 2 = 1 : 1.73205... : 2

Area = (a² × √3) ÷ 2

Perimeter = a(3 + √3)

Example: short leg = 6 → long leg = 6√3 = 10.392, hyp = 12, area = (36 × √3) ÷ 2 = 31.177

📖 How to Use This Calculator

Steps to Solve a Special Right Triangle

Choose a triangle type - Click the 45-45-90 tab for an isosceles right triangle, or 30-60-90 for the half-equilateral triangle.

Select which side you know - For the 45-45-90 triangle choose Leg or Hypotenuse. For the 30-60-90 triangle choose Short Leg (opposite 30°), Long Leg (opposite 60°), or Hypotenuse.

Enter the length - Type any positive number. The unit can be anything (cm, m, inches, feet) and the results are in the same unit.

Click Calculate - All sides, the area, the perimeter, and the exact side ratio appear instantly. Use the share buttons to copy the result or send via WhatsApp.

💡 Example Calculations

Example 1 - 45-45-90 Triangle from the Leg

A square room is 10 meters on each side. Find the diagonal.

The diagonal of a square divides it into a 45-45-90 triangle with leg = 10 m.

Hypotenuse = leg times root2 = 10 times 1.41421 = 14.1421 m.

Area = leg squared divided by 2 = 100 divided by 2 = 50 m². Perimeter = 2 times 10 + 14.142 = 34.142 m.

Hypotenuse = 14.142 m

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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.

Short leg (rise, opposite 30°) = hypotenuse divided by 2 = 8 divided by 2 = 4 feet.

Long leg (run, opposite 60°) = short leg times root3 = 4 times 1.73205 = 6.928 feet.

Area = 0.5 times 4 times 6.928 = 13.856 sq ft. Perimeter = 4 + 6.928 + 8 = 18.928 ft.

Short Leg = 4 ft, Long Leg = 6.928 ft

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Example 3 - 45-45-90 Triangle from the Hypotenuse

A fence gate diagonal brace is 12 feet long. Find the frame width and height.

This is a 45-45-90 triangle where the hypotenuse (brace) = 12 ft.

Leg = hypotenuse divided by root2 = 12 divided by 1.41421 = 8.485 feet.

Both the width and height of the gate equal 8.485 feet. Area = 8.485 squared divided by 2 = 35.999 square feet.

Each Leg (Width = Height) = 8.485 ft

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❓ Frequently Asked Questions

What makes a right triangle "special"?+

A right triangle is called special when its angles produce exact, simple side ratios. The 45-45-90 triangle has sides in ratio 1:1:root2, and the 30-60-90 triangle has sides in ratio 1:root3:2. Because these ratios are fixed, you only need one side to find all three, without using the Pythagorean theorem every time.

What is the hypotenuse of a 45-45-90 triangle with legs of length 5?+

The hypotenuse equals the leg multiplied by root2. So: 5 times root2 = 5 times 1.41421 = 7.071. The area is 5 squared divided by 2 = 12.5 square units, and the perimeter is 5 + 5 + 7.071 = 17.071 units.

What is the long leg of a 30-60-90 triangle with short leg 8?+

The long leg equals the short leg multiplied by root3. So: 8 times root3 = 8 times 1.73205 = 13.856. The hypotenuse is 2 times 8 = 16. The area is 0.5 times 8 times 13.856 = 55.425 square units.

How do you find all sides of a 30-60-90 triangle if you only know the hypotenuse?+

Divide the hypotenuse by 2 to get the short leg. Multiply the short leg by root3 to get the long leg. For example, with hypotenuse = 20: short leg = 10, long leg = 10 times root3 = 17.321.

What is the difference between a 45-45-90 and a 30-60-90 triangle?+

A 45-45-90 triangle is isosceles (two equal legs) formed by halving a square diagonally. A 30-60-90 triangle is scalene (all sides different lengths) formed by halving an equilateral triangle. The 45-45-90 uses the ratio 1:1:root2; the 30-60-90 uses 1:root3:2.

Can special right triangles have integer sides?+

No. Both special right triangles have irrational side ratios (root2 and root3 are irrational), so there is no way to have all three sides as whole numbers. Pythagorean triples like 3-4-5 and 5-12-13 are right triangles with integer sides, but their angles are not 45, 30, or 60 degrees.

How are special right triangles used on the SAT and ACT?+

Both exams include geometry problems where a triangle has angles 45-45-90 or 30-60-90. Memorizing the ratios 1:1:root2 and 1:root3:2 lets you solve these instantly. The SAT includes a reference sheet with these ratios, but recognizing them quickly saves significant time during the test.

Why is the 45-45-90 triangle called an isosceles right triangle?+

It is isosceles because two of its angles are equal (both 45 degrees), and isosceles triangles have two equal sides. The two legs of a 45-45-90 triangle are always the same length. It is also a right triangle because one angle is exactly 90 degrees.

What are the trig values for 30, 45, and 60 degrees?+

From 45-45-90: sin(45°) = cos(45°) = root2 divided by 2 = approximately 0.7071, tan(45°) = 1. From 30-60-90: sin(30°) = 0.5, cos(30°) = root3 divided by 2 = approximately 0.866, tan(30°) = 1 divided by root3. sin(60°) = root3 divided by 2, cos(60°) = 0.5, tan(60°) = root3 = approximately 1.732.

How do I find the area of a 30-60-90 triangle from the long leg?+

First find the short leg: short leg = long leg divided by root3. Then calculate area = 0.5 times short leg times long leg = long leg squared divided by (2 times root3) = long leg squared times root3 divided by 6. For a long leg of 12: short leg = 12 divided by root3 = 6.928, area = 0.5 times 6.928 times 12 = 41.569 square units.

Where do 30-60-90 triangles appear in architecture?+

The 30-60-90 triangle appears in hexagonal building layouts, structural roof trusses with a 30-degree pitch, staircase design (rise-run relationships), and geodesic dome paneling. Equilateral tiled floors and ceilings inherently contain 30-60-90 triangles in their geometry.

What is the perimeter of a 45-45-90 triangle with hypotenuse 10?+

First find the legs: leg = hypotenuse divided by root2 = 10 divided by 1.41421 = 7.071. Perimeter = 7.071 + 7.071 + 10 = 24.142 units. Area = leg squared divided by 2 = 7.071 squared divided by 2 = 50 divided by 2 = 25 square units.

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📌 Quick Tips

💡The 45-45-90 triangle has legs in ratio 1:1 and hypotenuse equal to leg times root 2 (approximately 1.414 times the leg).

💡The 30-60-90 triangle has sides in ratio 1 : root3 : 2. The short leg is half the hypotenuse.

💡To find the hypotenuse of a 45-45-90 triangle, multiply the leg by 1.41421 (root 2). For a 30-60-90 triangle, double the short leg.

💡Both special right triangles appear on the SAT, ACT, and most geometry exams. Memorize the ratios to save time on standardized tests.

💡The 45-45-90 triangle is half of a square cut diagonally. The 30-60-90 triangle is half of an equilateral triangle cut by an altitude.