45-45-90 Triangle: The Isosceles Right Triangle and Its Properties
Calculate all sides, area, and perimeter of a 45-45-90 isosceles right triangle. Learn the 1 : 1 : √2 ratio and solve problems with our free calculator.
The 45-45-90 triangle — also known as the isosceles right triangle — is the simplest of the special right triangles. With two equal angles of 45° and two equal legs, its sides follow the elegant ratio 1 : 1 : √2. This triangle is half of a square (formed by drawing a diagonal), making it fundamental to understanding the relationship between squares, their diagonals, and the Pythagorean theorem. From picture frame construction to vector decomposition in physics, the 45-45-90 triangle is everywhere.
Key Takeaway
In a 45-45-90 triangle, the legs are equal and the hypotenuse equals the leg times √2. If the leg is a, then the hypotenuse is a√2 and the area is a²/2. This triangle is exactly half of a square divided by its diagonal.
1. Deriving the 1 : 1 : √2 Ratio
The 45-45-90 triangle arises from bisecting a square along its diagonal. A square with side length a has a diagonal of length a√2 (by the Pythagorean theorem: d² = a² + a² = 2a). The diagonal divides the square into two congruent isosceles right triangles, each with legs of length a and hypotenuse of length a√2.
Hypotenuse: a√2
Area: a²/2
Perimeter: a(2 + √2)
2. Solving for Any Known Side
- From leg a: Hypotenuse = a√2, Area = a²/2
- From hypotenuse h: Leg = h/√2 = h√2/2, Area = h²/4
Worked Example
A 45-45-90 triangle has a leg of length 10. Find all properties.
- Leg = 10
- Hypotenuse = 10√2 ≈ 14.142
- Area = 10²/2 = 50
- Perimeter = 10 + 10 + 10√2 = 20 + 10√2 ≈ 34.142
3. Trigonometric Values
The 45-45-90 triangle gives us the exact trigonometric values for 45°:
| Function | 45° Value |
|---|---|
| sin 45° | √2/2 ≈ 0.7071 |
| cos 45° | √2/2 ≈ 0.7071 |
| tan 45° | 1 |
The equality sin 45° = cos 45° reflects the fact that the two legs are equal. The tangent equals 1 because opposite and adjacent sides are the same length.
4. Applications
The 45-45-90 triangle appears in roof pitch calculations (a 12:12 roof slope forms 45° angles), miter joint construction (cutting wood at 45° for picture frames), diagonal bracing in construction, vector decomposition when a force acts at 45° to the axes, and the geometry of the unit circle at the 45° position.
5. Frequently Asked Questions
Is a 45-45-90 triangle the same as an isosceles right triangle?
Yes. An isosceles right triangle has two equal legs and one 90° angle, which forces the angles to be 45°, 45°, and 90°. The two terms are interchangeable.
What is the altitude to the hypotenuse?
The altitude from the right angle to the hypotenuse equals a/√2 = a√2/2. It is also the median and angle bisector from the right angle.
Using Our Calculator
Select whether you know the leg or hypotenuse, enter the value, and the calculator instantly finds all remaining sides, area, and perimeter using the 1 : 1 : √2 ratio.