Fold a Square Corner to Corner — and the Triangle Appears
You've drawn this shape in grade school. Take a square piece of paper and fold it diagonally, corner to corner. The crease you just made is the hypotenuse of a 45-45-90 triangle. Both halves are identical right triangles with angles of 45°, 45°, and 90°.
That's not a coincidence. It's geometry doing what it always does: hiding useful structure in the most familiar objects.
The 45-45-90 triangle is an isosceles right triangle. Because both legs are equal, the ratio of sides isn't complicated to memorize — it falls straight out of the Pythagorean theorem.
The 1:1:√2 Ratio Comes From One Calculation
Start with a square of side length . The diagonal connects opposite corners. By the Pythagorean theorem:
Divide each side by and the ratio emerges:
That's it. If you know one leg, multiply by to get the hypotenuse. If you know the hypotenuse, divide by (or equivalently, multiply by ) to get each leg.
The move to memorize: one leg → multiply by √2 → hypotenuse. That's the whole triangle.
Three Examples, One Pattern
Once you see the multiplication pattern, solving these triangles is almost automatic.
Leg = 5
Hypotenuse =
Hypotenuse = 12
Each leg =
Leg = 3√2
Hypotenuse =
That last example trips people up. When the leg already contains a √2, the hypotenuse simplifies cleanly to a whole number. Worth watching for on tests.
Every Square Diagonal, Every Corner Cut, Every 45° Angle
This triangle doesn't stay confined to geometry homework.
A standard 12-inch floor tile, laid on a diagonal, occupies inches from corner to corner across the room. An installer calculating how many tiles fit on a diagonal grid needs that number.
TV screen sizes are measured diagonally. A 55-inch screen with a 16:9 aspect ratio has a width of about 47.9 inches and height of 27 inches — but the diagonal is exactly . That diagonal calculation is the Pythagorean theorem with a 45-45-90 built into the rounding assumptions when aspect ratios are close to square.
Carpentry is full of 45° miter cuts. Picture frame corners. Cabinet joints. Crown molding. Any time two pieces meet at a corner, the cut angle is 45° and the piece lengths follow the relationship.
45-45-90 vs. 30-60-90: Know Which One You Have
Both are "special right triangles," meaning you can solve them from one side without trig tables. But they come from different parent shapes and the ratios are different.
| Property | 45-45-90 | 30-60-90 |
|---|---|---|
| Parent shape | Square (folded diagonally) | Equilateral triangle (bisected) |
| Side ratio | 1 : 1 : √2 | 1 : √3 : 2 |
| Legs equal? | Yes (isosceles) | No (scalene) |
| Key multiplier | × √2 | × √3 (for long leg), × 2 (for hypotenuse) |
The 30-60-90 article walks through the equilateral bisection proof in detail. The core idea is the same: derive the ratio once, then reuse it everywhere.
Quick Questions
Why does the hypotenuse equal the leg times √2?
Because both legs are equal. The Pythagorean theorem gives , so . The √2 comes directly from having two identical legs.
Should I leave the answer as √2 or convert to a decimal?
Leave it as an exact radical unless the problem asks for a decimal or a specific number of significant figures. Exact form is cleaner and shows you understand the structure. Rounding to 1.414 introduces error that compounds in further calculations.
What if I'm given the hypotenuse and need the legs?
Divide the hypotenuse by √2, or multiply by (same thing, rationalized). If hypotenuse = 10, each leg = .
Where do 45-45-90 triangles appear in standardized tests?
SAT and ACT geometry sections frequently place 45-45-90 triangles inside squares, rhombuses, and coordinate grids. Recognize that any square diagonal creates one, and any 45° angle at a right angle creates one. The ratio eliminates the need to use trig functions.