🔺 45-45-90 Triangle Calculator
Solve special right triangles with sides in ratio 1:1:√2
How to Use This Calculator
Select Which Side You Know
Choose whether you know the leg (one of the equal sides) or the hypotenuse.
Enter the Length
Input the length of the side you know. Make sure to enter a positive number.
Calculate
Click the "Calculate" button to find all remaining sides, perimeter, and area.
Review Results
See both legs (equal), hypotenuse, angles (45°, 45°, 90°), perimeter, and area displayed clearly.
Formula
Ratio: 1 : 1 : √2
If leg = a, then: a : a : a√2
Given Leg (a):
- Leg 1 = a
- Leg 2 = a (equal to leg 1)
- Hypotenuse = a × √2
Given Hypotenuse (a√2):
- Leg 1 = Hypotenuse ÷ √2
- Leg 2 = Leg 1 (both legs are equal)
Area:
Area = (Leg × Leg) / 2 = a² / 2
Perimeter:
Perimeter = a + a + a√2 = 2a + a√2 = a(2 + √2)
Pythagorean Theorem:
Since legs are equal: a² + a² = (a√2)²
2a² = 2a² ✓
About 45-45-90 Triangle Calculator
The 45-45-90 triangle is a special right triangle (also called an isosceles right triangle) with angles measuring 45°, 45°, and 90°. Its sides are always in the ratio 1:1:√2, where the two legs are equal in length and the hypotenuse is √2 times the leg length.
When to Use This Calculator
- Geometry: Solve problems involving 45-45-90 triangles
- Trigonometry: Understand special angle relationships
- Construction: Calculate measurements for square-based structures (diagonals)
- Education: Learn about special right triangles
- Engineering: Design calculations requiring exact ratios
- Square Diagonals: Find diagonal length from side length or vice versa
Why Use Our Calculator?
- ✅ Quick Calculations: Find all sides instantly from one known side
- ✅ Accurate Results: Uses exact mathematical ratios
- ✅ Complete Information: Shows all sides, angles, area, and perimeter
- ✅ Flexible Input: Works with leg or hypotenuse
- ✅ Educational: Helps understand special triangle properties
- ✅ Free: No registration required
Key Properties
- Side Ratio: The sides are always in the ratio 1:1:√2
- Angles: Fixed at 45°, 45°, and 90°
- Isosceles: Both legs are equal in length
- Half Square: A 45-45-90 triangle is half of a square (cut along the diagonal)
- Right Triangle: Contains a 90° angle, making it a right triangle
Example
If one leg is 10 units:
- Leg 1 (opposite 45°) = 10
- Leg 2 (opposite 45°) = 10
- Hypotenuse (opposite 90°) = 10 × √2 ≈ 14.142
- Area = (10 × 10) / 2 = 50
- Perimeter = 10 + 10 + 14.142 ≈ 34.142
Relationship to Squares
A 45-45-90 triangle is exactly half of a square cut along its diagonal. If a square has side length s, its diagonal is s√2. The two resulting triangles are 45-45-90 triangles with legs of length s and hypotenuse of length s√2.
Frequently Asked Questions
What is a 45-45-90 triangle?
A 45-45-90 triangle is a special right triangle (isosceles right triangle) with angles of 45°, 45°, and 90°. Its sides are always in the ratio 1:1:√2, where both legs are equal.
What is the side ratio of a 45-45-90 triangle?
The sides are in the ratio 1:1:√2. If the leg is a, then both legs are a and the hypotenuse is a√2.
How do I find the hypotenuse of a 45-45-90 triangle?
The hypotenuse is always √2 times the leg length. If you know a leg is a, the hypotenuse is a√2.
Are the legs always equal in a 45-45-90 triangle?
Yes! In a 45-45-90 triangle, the two legs are always equal in length. This makes it an isosceles right triangle.
What's the relationship between 45-45-90 triangles and squares?
A 45-45-90 triangle is exactly half of a square cut along its diagonal. The diagonal of a square with side s is s√2, which matches the hypotenuse of the resulting triangles.
How do I find the leg from the hypotenuse?
Divide the hypotenuse by √2. If hypotenuse = h, then leg = h / √2 = h√2 / 2.
Can I use this calculator for triangles that aren't 45-45-90?
No, this calculator is specifically designed for 45-45-90 triangles. If your triangle doesn't have these exact angles, use a general triangle calculator instead.