45-45-90 Triangle
Calculate the sides, area, and perimeter of a 45-45-90 special right triangle.
45-45-90 — The Isosceles Right Triangle
Half a square. Ratio 1:1:√2. The diagonal of any square creates two perfect 45-45-90 triangles.
📐 Common 45-45-90 Examples — Click to Load
I Know the:
45-45-90 Triangle Visualization
Side Lengths
Side Proportions (1:1:√2)
Properties Radar
Step-by-Step Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- The 45-45-90 triangle has a fixed 1:1:√2 side ratio (leg : leg : hypotenuse)
- It is an isosceles right triangle — two equal legs and one 90° angle
- The diagonal of a square divides it into two congruent 45-45-90 triangles
- Given one side, all other sides and area can be found using the ratio — no trigonometry needed
- sin(45°) = cos(45°) = √2/2 and tan(45°) = 1 — the simplest trigonometric values
Did You Know?
How 45-45-90 Triangles Work
The 45-45-90 triangle derives from a square. Draw the diagonal of a square with side 1; by the Pythagorean theorem, diagonal = √2. Each half is a 45-45-90 triangle.
Given Leg → Hypotenuse
If leg = L, then hypotenuse = L × √2. Example: leg = 10 → hypotenuse ≈ 14.14.
Given Hypotenuse → Leg
If hypotenuse = H, then leg = H ÷ √2. Example: hypotenuse = 14.14 → leg ≈ 10.
Area & Perimeter
Area = leg²/2. Perimeter = 2×leg + leg√2 = leg(2 + √2).
Expert Tips
Memorize √2 ≈ 1.414
Hypotenuse ≈ 1.41 × leg. Leg ≈ hypotenuse ÷ 1.41.
Don't Confuse with 30-60-90
30-60-90 has ratio 1:√3:2. In 45-45-90, both legs are equal.
Square Diagonal Shortcut
Diagonal of square with side s = s√2. Side = diagonal ÷ √2.
Rationalize When Needed
leg = hypotenuse/√2 = hypotenuse×√2/2.
45-45-90 vs 30-60-90 vs General Right
| Property | 45-45-90 | 30-60-90 | General |
|---|---|---|---|
| Side ratio | 1:1:√2 | 1:√3:2 | Varies |
| Legs equal? | Yes | No | Maybe |
| Angles | 45°,45°,90° | 30°,60°,90° | Varies |
| From square diagonal? | Yes | No | No |
FAQ
What is the 45-45-90 triangle ratio?
leg : leg : hypotenuse = 1 : 1 : √2. If legs = 1, hypotenuse = √2 ≈ 1.414.
Why is it called isosceles right?
Isosceles = two equal sides (legs). Right = one 90° angle. The two equal angles must be 45° each.
How does the square diagonal relate?
A square's diagonal splits it into two 45-45-90 triangles. Diagonal = side × √2.
How do I find leg from hypotenuse?
leg = hypotenuse ÷ √2 = hypotenuse × √2/2. Or multiply by 0.7071 for a quick estimate.
What is the area formula?
Area = ½ × leg × leg = leg²/2. Half the area of a square with that side.
When to use in real life?
Construction (miter joints), design (45° rotations), navigation (equal N/E distances), engineering.
How to verify 45-45-90?
Check: 90° angle, two equal legs, hypotenuse = leg × √2.
Why is √2 irrational?
√2 cannot be written as a fraction. The Greeks proved this — a major discovery.
45-45-90 by the Numbers
Official Sources
Disclaimer: Results use standard floating-point precision. For critical construction or engineering, verify with professional tools.