What Is a 45 45 90 Triangle?
What the 45 45 90 Triangle Calculator Does
A 45 45 90 Triangle Calculator helps solve a special type of isosceles triangle that also forms a right angle. This triangle has two equal angles of 45° and one 90° angle. Because of its equal angles, the two shorter sides are equal in length. The calculator quickly finds the missing sides when you enter just one known value.
This tool makes solving geometry problems easier and faster by instantly applying the triangle’s fixed side ratio. It ensures accurate results and helps students and professionals quickly determine unknown side lengths without lengthy manual calculations.
The 1 : 1 : √2 Side Ratio
The 45 45 90 triangle follows a fixed side ratio (1 : 1 : √2).
- The two equal legs have the same length.
- The hypotenuse is √2 times one leg.
- These fixed properties make calculations simple and predictable.
How to Calculate a 45 45 90 Triangle
The calculator uses known geometry formulas and applies the fixed side ratio. When you enter one side, it calculates the other two automatically.
It may also confirm results using the Pythagorean theorem to ensure correct mathematical calculations, providing accurate answers for students and professionals.
Why This Triangle Matters in Geometry
The 45 45 90 triangle is common in geometry and trigonometry problems. It helps students understand trigonometric ratios such as sine, cosine, and tangent in a simple way. This triangle is closely related to other special right triangles, and comparison problems may also involve tools like the 30 60 90 Triangle Calculator..
The 45 45 90 Triangle Formula
The Standard 1 : 1 : √2 Ratio
A 45 45 90 triangle always follows this fixed side ratio:
- First leg = 1
- Second leg = 1
- Hypotenuse = √2
Because the legs are equal, this triangle is both a right triangle and an isosceles triangle.
How to Find the Hypotenuse
If one leg is known, the hypotenuse can be found using:
formula
Hypotenuse = Leg × √2
If one leg is 6 units: Hypotenuse = 6 × √2 ≈ 8.49 units. This follows directly from the fixed side ratio.
How to Find the Legs From the Hypotenuse
If the hypotenuse is known:
formula
Leg = Hypotenuse ÷ √2
If the hypotenuse is 10 units: Leg = 10 ÷ √2 ≈ 7.07 units. Both legs will have the same value because of equal angles.
Checking It With the Pythagorean Theorem
The 45 45 90 triangle follows the Pythagorean theorem:
formula
(Leg)² + (Leg)² = (Hypotenuse)²
Since both legs are equal: 2 × (Leg)² = (Hypotenuse)². This confirms the √2 relationship in a clear mathematical way.
A 45 45 90 Triangle Example, Step by Step
Assume one leg is 5 units.
- Hypotenuse = 5 × √2 ≈ 7.07
- Other leg = 5 (equal legs)
This simple step-by-step solution shows how easy calculations become with fixed triangle properties.
How the Calculator Applies These Rules
The calculator takes one known side and applies the fixed ratio automatically. It calculates the missing sides instantly and verifies results using geometry formulas.
This removes manual errors and helps students solve right triangle problems quickly.
45 45 90 Triangle Chart: Side Ratios
The isosceles right triangle — half a square cut along its diagonal — keeps its sides in the ratio 1 : 1 : √2:
| Side | Ratio | If each leg = 5 |
|---|---|---|
| Leg (45°) | 1 | 5 |
| Leg (45°) | 1 | 5 |
| Hypotenuse (90°) | √2 ≈ 1.414 | ≈ 7.07 |
Both directions in one line: hypotenuse = leg × √2, and leg = hypotenuse ÷ √2 (equivalently × √2/2 ≈ 0.707). This is also why a square's diagonal is always about 41% longer than its side — the same √2 at work.