45-45-90 Triangle Calculator
Solve 45°-45°-90° isosceles right triangles from any known leg, hypotenuse, area, or altitude. Generates exact radical expressions ($1 : 1 : \sqrt{2}$), algebraic proofs, and proportional SVG vector geometry.
45-45-90 Triangle Identity
A 45-45-90 triangle is an isosceles right triangle with two congruent interior angles of 45° and one 90° right angle. Its side lengths always satisfy the exact ratio 1 : 1 : √2 (leg : leg : hypotenuse).
Geometric Foundations & Square Diagonal Proof
The 45°-45°-90° triangle is the unique right triangle that is also isosceles. It is created naturally whenever a square is divided along its diagonal.
Formal Algebraic Proof of the $1 : 1 : \sqrt{2}$ Ratio
- Construct a Geometric Square: Consider a square $ABCD$ with side length $a$. All four corner angles measure $90^\circ$.
- Draw the Diagonal: Draw diagonal segment $AC$. Because the square has equal side lengths ($AB = BC = a$), $\triangle ABC$ is an isosceles right triangle.
- Angle Calculation: The diagonal bisects the $90^\circ$ right angles at vertices $A$ and $C$, creating two congruent acute angles of $45^\circ$ ($\angle BAC = \angle BCA = 45^\circ$).
- Apply the Pythagorean Theorem: Let $c$ represent the hypotenuse length $AC$:
$a^2 + a^2 = c^2 \implies 2a^2 = c^2 \implies c = \sqrt{2a^2} = a\sqrt{2}$
- Conclusion: The sides of any 45°-45°-90° triangle are in the exact proportion $a : a : a\sqrt{2}$, confirming the canonical $1 : 1 : \sqrt{2}$ identity.
Exact Ratio Matrix & Geometric Properties
Because the two legs are congruent ($a = b$), every linear and area property of a 45-45-90 triangle can be expressed as a function of the single leg parameter $a$.
| Property | Symbol | Exact Formula (in terms of leg $a$) | Numerical Value ($a = 1$) |
|---|---|---|---|
| Base Leg (opp 45°) | $a$ | $a$ | 1.0000 |
| Height Leg (opp 45°) | $b$ | $a$ | 1.0000 |
| Hypotenuse (opp 90°) | $c$ | $a\sqrt{2}$ | 1.4142 |
| Enclosed Area | $A$ | $\frac{1}{2} a^2$ | 0.5000 |
| Perimeter | $P$ | $2a + a\sqrt{2} = a(2 + \sqrt{2})$ | 3.4142 |
| Altitude to Hypotenuse | $h_c$ | $\frac{a\sqrt{2}}{2} = \frac{c}{2}$ | 0.7071 |
| Inradius | $r$ | $\frac{2a - a\sqrt{2}}{2} = \frac{a(2 - \sqrt{2})}{2}$ | 0.2929 |
| Circumradius | $R$ | $\frac{c}{2} = \frac{a\sqrt{2}}{2}$ | 0.7071 |
Step-by-Step Solving Algorithms for Every Known Input
Direct Radical Multiplication
- 1. Other Leg $b = a$
- 2. Hypotenuse $c = a\sqrt{2}$
- 3. $\text{Area} = \frac{a^2}{2}$
Rationalize the Denominator
- 1. $a = b = \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}$
- 2. $\text{Area} = \frac{c^2}{4}$
- 3. Altitude $h_c = \frac{c}{2}$
Square Root Inversion
- 1. $a = b = \sqrt{2A}$
- 2. $c = \sqrt{2A} \cdot \sqrt{2} = \sqrt{4A} = 2\sqrt{A}$
- 3. Perimeter $P = 2\sqrt{2A} + 2\sqrt{A}$
Hypotenuse Bisection
- 1. Hypotenuse $c = 2h_c$
- 2. $a = b = h_c\sqrt{2}$
- 3. $\text{Area} = h_c^2$
Exact Trigonometric Values and Unit Circle Coordinates
The 45-45-90 triangle is responsible for the symmetric unit circle coordinates in all four quadrants at angle increments of $\frac{\pi}{4}$ ($45^\circ, 135^\circ, 225^\circ, 315^\circ$).
Real-World Applications & Construction Geometry
Carpentry & Miter Cuts
Joining picture frames, baseboards, and crown molding at $90^\circ$ corners requires cutting boards at exact $45^\circ$ miter angles, forming 45-45-90 corner triangles.
Diagonal Square Bracing
Timber framing and scaffolding utilize $45^\circ$ diagonal cross-braces to prevent shear racking. The required brace length is calculated directly using $L = s\sqrt{2}$.
Screen Diagonals & Displays
Square display panels and camera sensor viewfinders use 45-45-90 calculations to convert diagonal screen specifications into horizontal and vertical pixel dimensions.
Graded Step-by-Step Numerical Solutions
Solve a 45-45-90 triangle with leg $a = 12\text{ in}$.
1. Find other leg $b$: $b = a = 12\text{ in}$.
2. Find hypotenuse $c$: $c = a\sqrt{2} = 12\sqrt{2} \approx 16.9706\text{ in}$.
3. Calculate Area: $\text{Area} = \frac{12 \times 12}{2} = 72\text{ sq in}$.
4. Calculate Perimeter: $P = 12 + 12 + 12\sqrt{2} = 24 + 12\sqrt{2} \approx 40.9706\text{ in}$.
Solve a 45-45-90 triangle with hypotenuse $c = 18\text{ cm}$.
1. Solve for legs $a = b$: $a = \frac{18}{\sqrt{2}} = \frac{18\sqrt{2}}{2} = 9\sqrt{2} \approx 12.7279\text{ cm}$.
2. Calculate Area: $\text{Area} = \frac{(9\sqrt{2})^2}{2} = \frac{81 \times 2}{2} = 81\text{ cm}^2$.
3. Calculate Altitude to Hypotenuse: $h_c = \frac{c}{2} = \frac{18}{2} = 9\text{ cm}$.
Common Pitfalls & Mistake Avoidance
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