Special right triangle tool
45-45-90 Triangle Calculator
Enter any one side of a 45-45-90 isosceles right triangle and instantly find all three sides, with the √2 ratio, the formula, and step-by-step working.
- Enter any side
- Step-by-step working
- Exact √2 form
- 100% free
Calculator
45-45-90 Triangle Side Calculator
For the general theorem behind the √2 relationship, see Pythagorean Theorem.
Ratio
The 45-45-90 Triangle Ratio
1 : 1 : √2
(leg a) : (leg b) : (hypotenuse)
A 45-45-90 triangle is an isosceles right triangle. It has a 90-degree angle and two equal
45-degree angles, so its two legs are equal. The hypotenuse is always the leg length multiplied
by √2, approximately 1.414. This fixed 1 : 1 : √2 ratio
means you only need one side to find all three.
The √2 relationship comes directly from the Pythagorean theorem: if both legs equal
a, then c² = a² + a² = 2a², so c = a√2. This is the most
symmetric special right triangle, and it appears naturally in square diagonals, drafting tools,
and equal-rise equal-run layouts.
| Known side | Leg a | Leg b | Hypotenuse (c) |
|---|---|---|---|
| Leg = 1 | 1 | 1 | √2 ≈ 1.414 |
| Leg = 5 | 5 | 5 | 5√2 ≈ 7.071 |
| Leg = 10 | 10 | 10 | 10√2 ≈ 14.142 |
| Hypotenuse = 1 | √2 / 2 ≈ 0.707 | √2 / 2 ≈ 0.707 | 1 |
| Hypotenuse = √2 | 1 | 1 | √2 ≈ 1.414 |
| Hypotenuse = 10 | 5√2 ≈ 7.071 | 5√2 ≈ 7.071 | 10 |
Formulas
45-45-90 Triangle Formulas
b = a
—
c = a√2
≈ 1.414a
A = a² / 2
—
P = a(2 + √2)
≈ 3.414a
a = b = c√2 / 2
≈ 0.707c
A = c² / 4
—
P = c(1 + √2)
≈ 2.414c
These formulas come directly from the 1 : 1 : √2 ratio. Once you know one side,
multiply or divide by the matching factor to find the others. The calculator above applies the same
rules automatically and keeps both the exact √2 form and the decimal approximation in view.
Where does the 1 : 1 : √2 ratio come from?
A 45-45-90 triangle is exactly half of a square, cut along the diagonal. If the square has
side length a, the diagonal has length a√2 by the Pythagorean
theorem: √(a² + a²) = a√2.
Cutting the square diagonally produces two identical 45-45-90 triangles, each with legs of
length a and hypotenuse a√2. That is why the ratio is
1 : 1 : √2 and why this triangle appears whenever you work with squares and their
diagonals.
Applications
Real-World Applications
The 45-45-90 triangle appears in any situation involving a square, a 45-degree angle, or an equal horizontal and vertical distance.
Diagonal of a Square
The diagonal of a square with side s is exactly s√2. The diagonal
splits the square into two 45-45-90 triangles.
Example: A square tile with side 30 cm has a diagonal of 30√2 ≈ 42.43 cm.
Equal Rise and Run
Any ramp, staircase, or slope where the vertical rise equals the horizontal run forms a
45-45-90 triangle. The slope length is the rise multiplied by √2.
Example: A ramp that rises 2 m over 2 m has a slope length of 2√2 ≈ 2.83 m.
45-Degree Set Square
The standard 45-degree set square used in drafting and technical drawing is a 45-45-90 triangle. It is used to draw 45° lines and construct clean perpendiculars.
Example: In technical drawing, the equal legs make mirrored 45° layouts easy to place.
Examples
Worked Examples
These three examples cover both input types, knowing a leg and knowing the hypotenuse, plus the classic square-diagonal scenario.
Example 1: Leg = 8
Given: a = 8. Find b and c.
| Leg b | b = a = 8 |
|---|---|
| Hypotenuse | c = 8√2 ≈ 11.314 |
| Area | A = 8² / 2 = 64 / 2 = 32 |
| Perimeter | P = 8 + 8 + 8√2 = 16 + 8√2 ≈ 27.314 |
Example 2: Hypotenuse = 6
Given: c = 6. Find a and b.
| Each leg | a = b = 6 / √2 = 6√2 / 2 = 3√2 ≈ 4.243 |
|---|---|
| Area | A = 6² / 4 = 36 / 4 = 9 |
| Perimeter | P = 3√2 + 3√2 + 6 = 6√2 + 6 ≈ 14.485 |
Example 3: A square room is 5 m × 5 m. How long is the diagonal?
The diagonal of the room forms a 45-45-90 triangle with both legs equal to 5 m.
| Hypotenuse | c = 5√2 ≈ 7.071 m |
|---|
Compare
45-45-90 vs 30-60-90 Triangle
| Feature | 45-45-90 | 30-60-90 |
|---|---|---|
| Angles | 45°, 45°, 90° | 30°, 60°, 90° |
| Side ratio | 1 : 1 : √2 | 1 : √3 : 2 |
| Legs | Equal (isosceles) | Unequal |
| Hypotenuse | leg × √2 | 2 × short leg |
| Origin | Half of a square | Half of an equilateral triangle |
| Common use | Square diagonals, 45° angles | Ramps, roofs, 30°/60° angles |
Both are special right triangles with fixed side ratios that can be solved from one known side. For 30-60-90 calculations, use the 30-60-90 Triangle Calculator.
FAQ
Frequently Asked Questions
The sides are in the ratio 1 : 1 : √2. Both legs are equal in length, and the
hypotenuse is the leg multiplied by √2, about 1.414. For example,
if each leg is 7, the hypotenuse is 7√2 ≈ 9.899.
Multiply the leg length by √2. If the leg is a, the hypotenuse is
a√2. For a leg of 5, the hypotenuse is 5√2 ≈ 7.071.
Divide the hypotenuse by √2, or multiply it by √2 / 2. If the
hypotenuse is 10, each leg is 10 / √2 = 5√2 ≈ 7.071.
An isosceles triangle has at least two equal sides. In a 45-45-90 triangle, the two legs are always equal because the two 45-degree angles opposite them are equal.
The area is a² / 2, where a is the leg length. If you know the
hypotenuse c instead, the area is c² / 4. For a leg of
6, the area is 36 / 2 = 18.
A 45-45-90 triangle is exactly half of a square, formed by cutting the square along one diagonal. The square's side becomes the triangle's leg, and the diagonal becomes the hypotenuse.
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